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 | #61289 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #532 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1251 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)(15,21)(19,29)(20,30)(22,34)(23,35)(24,33)(25,50)(26,41)(27,38)(28,54)(31,40)(36,49)(37,48)(39,55)(42,53)(43,52)(44,56)(45,67)(46,61)(47,57)(51,62)(58,68)(59,64)(60,65)(63,69)(66,71)(70,72);(1,3)(2,6)(4,7)(5,8)(9,11,15,23)(10,16,21,35)(12,22,26,46)(13,24,31,47)(14,25,30,48)(17,33,40,57)(18,34,41,61)(19,36,39,58)(20,37,32,50)(27,44,51,66)(28,45,52,65)(29,49,55,68)(38,56,62,71)(42,59,63,70)(43,60,54,67)(53,64,69,72);(1,4,5,2)(3,7,8,6)(10,17)(13,29)(14,30)(16,33)(18,38)(19,20)(21,40)(24,49)(25,48)(27,53)(28,52)(31,55)(32,39)(34,56)(35,57)(36,37)(41,62)(42,43)(44,64)(45,65)(47,68)(50,58)(51,69)(54,63)(59,60)(61,71)(66,72)(67,70);(3,8)(6,7)(9,12)(10,18)(11,22)(13,27)(14,28)(15,26)(16,34)(17,38)(19,42)(20,43)(21,41)(23,46)(24,44)(25,45)(29,53)(30,52)(31,51)(32,54)(33,56)(35,61)(36,59)(37,60)(39,63)(40,62)(47,66)(48,65)(49,64)(50,67)(55,69)(57,71)(58,70)(68,72);(11,23)(12,26)(16,35)(18,41)(24,47)(25,48)(27,51)(28,52)(33,57)(36,58)(37,50)(38,62)(42,63)(43,54)(49,68)(53,69)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,g^a=gh,h^a=hi,d^b=df,e^b=ei,g^c=gh,h^c=hi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531035);