Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623604 |
| 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 |
| prime radical | #532 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #532 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1248 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 72 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,16)(12,18)(13,17)(14,32)(15,19)(20,29)(21,30)(22,34)(23,37)(24,33)(25,50)(26,43)(27,38)(28,54)(31,39)(35,49)(36,48)(40,55)(41,53)(42,52)(44,56)(45,67)(46,59)(47,58)(51,63)(57,68)(60,64)(61,65)(62,69)(66,70)(71,72);(1,3)(2,6)(4,7)(5,8)(9,11,15,23)(10,16,19,37)(12,22,26,46)(13,24,31,47)(14,25,30,48)(17,33,39,58)(18,34,43,59)(20,35,40,57)(21,36,32,50)(27,44,51,66)(28,45,52,65)(29,49,55,68)(38,56,63,70)(41,60,62,71)(42,61,54,67)(53,64,69,72);(1,4,5,2)(3,7,8,6)(10,17)(13,29)(14,30)(16,33)(18,38)(19,39)(20,21)(24,49)(25,48)(27,53)(28,52)(31,55)(32,40)(34,56)(35,36)(37,58)(41,42)(43,63)(44,64)(45,65)(47,68)(50,57)(51,69)(54,62)(59,70)(60,61)(66,72)(67,71);(3,8)(6,7)(9,12)(10,18)(11,22)(13,27)(14,28)(15,26)(16,34)(17,38)(19,43)(20,41)(21,42)(23,46)(24,44)(25,45)(29,53)(30,52)(31,51)(32,54)(33,56)(35,60)(36,61)(37,59)(39,63)(40,62)(47,66)(48,65)(49,64)(50,67)(55,69)(57,71)(58,70)(68,72);(10,19)(11,23)(12,26)(17,39)(20,40)(21,32)(24,47)(25,48)(27,51)(28,52)(34,59)(49,68)(53,69)(56,70)(60,71)(61,67)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gh,h^a=hi,d^b=df,e^b=ei,g^c=gh,h^c=hi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531041);