Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623626 |
| 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 | #61477 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #529 ◁ #2966 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1257 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)(9,10,16,20)(11,17,25,43)(12,28,32,62)(13,22,15,19)(14,34,37,48)(18,44,47,31)(21,33,49,36)(23,52,29,61)(24,42,27,40)(26,57,56,68)(30,64,38,50)(35,51,66,46)(39,63,45,54)(41,55,70,59)(53,72,60,69)(58,71,65,67);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,29)(13,24)(14,26)(15,27)(16,25)(18,45)(19,40)(20,43)(21,41)(22,42)(23,32)(28,61)(30,60)(31,63)(33,55)(34,57)(35,65)(36,59)(37,56)(38,53)(39,47)(44,54)(46,71)(48,68)(49,70)(50,72)(51,67)(52,62)(58,66)(64,69);(9,12)(10,18)(11,23)(13,30)(14,35)(15,38)(16,32)(17,39)(19,46)(20,47)(21,50)(22,51)(24,53)(25,29)(26,58)(27,60)(28,34)(31,36)(33,44)(37,66)(40,67)(41,69)(42,71)(43,45)(48,62)(49,64)(52,57)(54,59)(55,63)(56,65)(61,68)(70,72);(2,5)(6,8)(9,13,16,15)(10,19,20,22)(11,24,25,27)(12,30,32,38)(14,33,37,36)(17,40,43,42)(18,46,47,51)(21,48,49,34)(23,53,29,60)(26,55,56,59)(28,50,62,64)(31,35,44,66)(39,67,45,71)(41,68,70,57)(52,69,61,72)(54,58,63,65);(1,3,4,7)(2,6,5,8)(10,20)(11,25)(18,47)(19,22)(21,49)(23,29)(24,27)(26,56)(28,62)(34,48)(46,51)(50,64)(53,60)(54,63)(55,59)(58,65)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,g^a=gh,h^a=hi,c^b=ci,d^b=df,e^b=ei,g^c=gh,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531263);