Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #35737 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1723 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18)(19,23)(20,24)(21,26)(22,25)(27,34)(28,35)(29,37)(30,36)(31,38)(32,39)(33,40)(41,45)(42,46)(43,47)(44,48);(2,6)(7,8)(9,11,12,16)(10,13,14,15)(17,19,22,30)(18,23,25,36)(20,27,31,42)(21,28,33,29)(24,34,38,46)(26,35,40,37)(32,43,44,41)(39,47,48,45);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,20)(18,24)(19,27)(21,32)(22,31)(23,34)(25,38)(26,39)(28,43)(29,41)(30,42)(33,44)(35,47)(36,46)(37,45)(40,48);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(19,28)(20,31)(21,33)(23,35)(24,38)(26,40)(27,41)(29,30)(34,45)(36,37)(42,43)(46,47);(2,6)(7,8)(10,14)(13,15)(18,25)(20,31)(23,36)(26,40)(27,42)(32,44)(35,37)(41,43)Polycyclic presentation:
2^9|b^2=gi,c^2=fg,d^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7476964);