Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #153597 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13654 |
| commutator | C32 | C22×C4 |
| Frattini, cosocle | C2×C32 | C23 |
| prime radical | #2262 | C22 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ C4×C16 ◁ #1397 ◁ G | |
| LECS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
111 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 111 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 128 points:
(1,2,6,12)(3,15)(4,22,25,82)(5,28,32,60)(7,36,35,53)(8,39,23,14)(9,41)(10,47,49,27)(11,50,54,33)(13,31,56,26)(16,64)(17,46)(18,76)(19,79)(20,80)(21,66)(24,86,30,97)(29,57)(34,102,84,93)(37,63)(38,107,85,83)(40,65)(42,111)(43,95)(44,71)(45,75)(48,117,52,123)(51,72)(55,101,109,119)(58,90,104,100)(59,110)(61,69)(62,73)(67,113)(68,122)(70,128)(74,114)(77,118)(78,121)(81,98,124,88)(87,126)(89,92,99,91)(94,115)(96,127)(103,112)(105,116)(106,120)(108,125);(1,3)(2,9)(4,16)(5,29)(6,21)(7,37)(8,40)(10,42)(11,51)(12,46)(13,57)(14,59)(15,60)(17,50)(18,26)(19,27)(20,23)(22,44)(24,87)(25,70)(28,66)(30,94)(31,78)(32,73)(33,41)(34,103)(35,72)(36,43)(38,108)(39,45)(47,69)(48,118)(49,114)(52,120)(53,116)(54,63)(55,125)(56,62)(58,127)(61,92)(64,97)(65,93)(67,98)(68,85)(71,88)(74,117)(75,101)(76,124)(77,100)(79,91)(80,102)(81,121)(82,113)(83,126)(84,96)(86,128)(89,95)(90,106)(99,105)(104,112)(107,115)(109,122)(110,119)(111,123);(1,4,23,47,6,25,8,27)(2,10,39,82,12,49,14,22)(3,16,20,69,21,70,40,19)(5,30,84,89,32,24,34,99)(7,26,85,100,35,31,38,90)(9,42,45,113,46,114,59,44)(11,52,109,81,54,48,55,124)(13,36,104,107,56,53,58,83)(15,61,65,64,66,79,80,128)(17,71,75,74,41,67,110,111)(18,68,77,72,78,108,106,37)(28,91,102,86,60,92,93,97)(29,94,96,95,73,87,103,105)(33,98,119,123,50,88,101,117)(43,112,115,62,116,127,126,57)(51,120,122,121,63,118,125,76)Polycyclic presentation:
2^9|a^2=e,c^2=eh,d^2=fgi,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ch,d^a=df,f^a=fg,g^a=gh,h^a=hi,d^b=dfi,e^b=ei,f^b=fg,g^b=gh,h^b=hi,d^c=di56 conjugacy classes — view the full character table.
| maximal subgroup of | #72321109, #72321113, #72321369, … |
| maximal normal subgroup of | #72321109, #72321113, #72321369, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60793);