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 | #121 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #529 ◁ #2966 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1254 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 72 points:
(2,5)(6,8)(9,10,13,18)(11,21,23,16)(12,26,27,62)(14,32,30,61)(15,35,31,20)(17,40,41,28)(19,44,43,34)(22,50,51,60)(24,56,54,65)(25,39,55,49)(29,63,36,45)(33,46,64,42)(37,52,47,69)(38,58,48,70)(53,67,59,71)(57,66,72,68);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,22)(13,23)(14,24)(15,25)(17,37)(18,21)(19,38)(20,39)(26,60)(27,51)(28,52)(29,53)(30,54)(31,55)(32,65)(33,57)(34,58)(35,49)(36,59)(40,69)(41,47)(42,66)(43,48)(44,70)(45,67)(46,68)(50,62)(56,61)(63,71)(64,72);(9,12,13,27)(10,17,18,41)(11,22,23,51)(14,33,30,64)(15,36,31,29)(16,37,21,47)(19,45,43,63)(20,46,35,42)(24,57,54,72)(25,59,55,53)(26,61,62,32)(28,44,40,34)(38,67,48,71)(39,68,49,66)(50,65,60,56)(52,70,69,58);(2,5)(6,8)(11,23)(16,21)(22,51)(24,54)(25,55)(37,47)(38,48)(39,49)(50,60)(52,69)(53,59)(56,65)(57,72)(58,70)(66,68)(67,71);(1,3,4,7)(2,6,5,8)(11,23)(16,21)(22,51)(24,54)(25,55)(37,47)(38,48)(39,49)(50,60)(52,69)(53,59)(56,65)(57,72)(58,70)(66,68)(67,71)Polycyclic presentation:
2^9|a^2=i,c^2=i,e^2=fi,g^2=hi,h^2=i,b^a=bf,c^a=cg,g^a=gh,h^a=hi,d^b=df,e^b=ei,g^c=gh,h^c=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531227);