Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150144 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2140 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C4×C16 | C23 |
| socle | C2 | #11661 |
| prime radical | #3414 | C4 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
220 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 66 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:
(1,2,6,13,8,14,39,61,9,16,40,62,43,15,7,41)(3,17,20,65,21,66,79,122,23,67,80,124,22,81,42,56)(4,24,27,83,29,55,28,53,30,54,38,84,69,11,49,52)(5,31,34,58,36,96,108,60,37,98,109,59,91,97,35,12)(10,32,45,76,47,100,115,78,48,102,116,77,63,101,46,19)(18,68,71,121,25,85,87,118,74,125,72,128,73,64,88,123)(26,89,92,51,94,95,93,82,33,103,105,57,107,119,106,120)(44,104,110,75,112,126,111,127,113,90,114,70,50,99,117,86);(1,3)(2,10)(4,25)(5,32)(6,20)(7,42)(8,21)(9,23)(11,50)(12,56)(13,45)(14,47)(15,63)(16,48)(17,31)(18,69)(19,35)(22,43)(24,44)(26,90)(27,87)(28,72)(29,74)(30,73)(33,104)(34,76)(36,100)(37,102)(38,88)(39,79)(40,80)(41,46)(49,71)(51,118)(52,117)(53,111)(54,113)(55,112)(57,123)(58,65)(59,124)(60,122)(61,115)(62,116)(64,103)(66,96)(67,98)(68,119)(70,92)(75,105)(77,109)(78,108)(81,97)(82,128)(83,110)(84,114)(85,89)(86,93)(91,101)(94,99)(95,125)(106,127)(107,126)(120,121);(1,4,8,29,9,30,43,69)(2,11,15,54,16,55,14,24)(3,18,22,73,23,74,21,25)(5,33,36,107,37,26,91,94)(6,38,39,49,40,27,7,28)(10,44,47,112,48,113,63,50)(12,57,59,82,60,51,58,120)(13,53,41,83,62,52,61,84)(17,64,66,68,67,85,81,125)(19,75,77,86,78,70,76,127)(20,72,42,87,80,71,79,88)(31,95,97,89,98,119,96,103)(32,99,101,90,102,126,100,104)(34,92,108,93,109,105,35,106)(45,114,115,117,116,110,46,111)(56,121,65,118,122,128,124,123)Polycyclic presentation:
2^9|a^2=e,c^2=h,d^2=fh,e^2=g,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=chi,d^b=df,f^b=fh,h^b=hi,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72320449 |
| maximal normal subgroup of | #72320449 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57341);