Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152901 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13206 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | C2×#508 | C22 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
306 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 306 conjugacy classes of subgroups — too many to draw. Only its 92 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)(3,16)(4,23,27,77)(5,30)(6,13)(7,38)(8,40)(9,15)(10,19)(11,26,49,29)(12,33)(14,31)(17,52)(18,64,67,107)(20,56)(21,73)(22,59)(24,79,82,54)(25,84,87,101)(28,91,90,50)(32,93)(34,96)(35,41)(36,58)(37,53)(39,98)(42,81,75,122)(43,70)(44,105)(45,95)(46,72)(47,66,102,119)(48,65,109,116)(51,63)(55,71)(57,69)(60,104)(61,114,80,76)(62,94)(68,110,120,126)(74,111)(78,89,124,85)(83,117,128,121)(86,100,88,92)(97,99)(103,112,123,118)(106,113)(108,127,125,115);(1,3,17,60,7,21,34,70,6,36,62,105,37,97,5,19,9,22,33,71,39,74,32,95,8,20,63,113,31,69,35,72)(2,10,41,99,14,44,51,58,40,43,93,73,98,104,12,16,15,46,30,57,53,106,94,56,13,45,96,111,38,55,52,59)(4,24,80,127,28,83,86,66,26,18,65,118,89,68,87,122,27,82,61,115,90,128,88,119,29,67,116,112,85,120,25,81)(11,47,92,121,50,108,114,54,77,42,101,110,124,103,109,64,49,102,100,117,91,125,76,79,23,75,84,126,78,123,48,107);(1,4)(2,11,15,49)(3,18,22,67)(5,25)(6,26)(7,28)(8,29)(9,27)(10,42)(12,48,52,109)(13,23,40,77)(14,50,53,91)(16,54)(17,61)(19,66,72,119)(20,24,36,82)(21,68,74,120)(30,92,41,100)(31,85)(32,86)(33,80)(34,88)(35,87)(37,89)(38,78,98,124)(39,90)(43,102)(44,103)(45,47)(46,75)(51,76,94,114)(55,108)(56,64)(57,110)(58,107)(59,79)(60,112,71,118)(62,116)(63,65)(69,83,97,128)(70,81,95,122)(73,121)(84,96,101,93)(99,126)(104,125)(105,127,113,115)(106,123)(111,117)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=eh,d^2=gh,e^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,c^b=ch,e^b=ei,f^c=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60098);