Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624150 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #525 ◁ #2929 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1499 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,17)(12,27)(13,19)(14,32)(15,36)(16,21)(18,29)(20,34)(22,47)(23,39)(24,40)(25,53)(26,56)(28,59)(30,60)(31,46)(33,42)(35,58)(37,45)(38,50)(41,55)(43,62)(44,61)(48,69)(49,70)(51,66)(52,68)(54,63)(57,67)(64,71)(65,72);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,22)(13,23)(14,25)(15,26)(16,24)(18,38)(19,39)(20,41)(21,40)(27,47)(28,48)(29,50)(30,49)(31,51)(32,53)(33,54)(34,55)(35,52)(36,56)(37,57)(42,63)(43,65)(44,64)(45,67)(46,66)(58,68)(59,69)(60,70)(61,71)(62,72);(9,12,16,30)(10,18,21,44)(11,22,24,49)(13,28,15,37)(14,33,35,62)(17,38,40,64)(19,42,36,43)(20,45,46,59)(23,48,26,57)(25,54,52,72)(27,58,60,32)(29,31,61,34)(39,63,56,65)(41,67,66,69)(47,68,70,53)(50,51,71,55);(3,8)(5,7)(9,13,16,15)(10,19,21,36)(11,23,24,26)(12,28,30,37)(14,31,35,34)(17,39,40,56)(18,42,44,43)(20,32,46,58)(22,48,49,57)(25,51,52,55)(27,59,60,45)(29,33,61,62)(38,63,64,65)(41,53,66,68)(47,69,70,67)(50,54,71,72);(2,6)(5,7)(11,24)(17,40)(22,49)(23,26)(25,52)(38,64)(39,56)(41,66)(47,70)(48,57)(50,71)(51,55)(53,68)(54,72)(63,65)(67,69)Polycyclic presentation:
2^9|c^2=i,d^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,d^b=df,e^b=ei,g^c=gh,h^c=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531352);