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 | #33579 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #533 ◁ #2960 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1264 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)(6,7)(9,10,14,20)(11,23,25,17)(12,28,30,62)(13,22,16,19)(15,35,34,46)(18,42,44,31)(21,33,47,37)(24,52,53,61)(26,40,32,45)(27,59,57,66)(29,63,38,48)(36,49,65,43)(39,55,50,70)(41,58,51,64)(54,71,56,69)(60,68,72,67);(1,2,4,5)(3,7,8,6)(9,11,14,25)(10,17,20,23)(12,24,30,53)(13,32,16,26)(15,27,34,57)(18,39,44,50)(19,45,22,40)(21,41,47,51)(28,61,62,52)(29,56,38,54)(31,55,42,70)(33,64,37,58)(35,66,46,59)(36,60,65,72)(43,68,49,67)(48,69,63,71);(9,12)(10,18)(11,24)(13,29)(14,30)(15,36)(16,38)(17,39)(19,43)(20,44)(21,48)(22,49)(23,50)(25,53)(26,54)(27,60)(28,35)(31,37)(32,56)(33,42)(34,65)(40,67)(41,69)(45,68)(46,62)(47,63)(51,71)(52,59)(55,58)(57,72)(61,66)(64,70);(1,3,4,8)(2,6,5,7)(11,25)(17,23)(24,53)(26,32)(27,57)(39,50)(40,45)(41,51)(52,61)(54,56)(55,70)(58,64)(59,66)(60,72)(67,68)(69,71);(9,13,14,16)(10,19,20,22)(11,26,25,32)(12,29,30,38)(15,33,34,37)(17,40,23,45)(18,43,44,49)(21,46,47,35)(24,54,53,56)(27,58,57,64)(28,48,62,63)(31,36,42,65)(39,67,50,68)(41,66,51,59)(52,69,61,71)(55,72,70,60)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=fi,e^2=i,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=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531105);