Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | #25131 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1667 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,29)(20,25)(21,35)(22,27)(23,28)(24,32)(26,39)(30,47)(31,41)(33,45)(34,43)(36,44)(37,42)(38,46)(40,48);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,32)(18,24,27,29)(20,30,31,48)(21,36,39,34)(23,37,38,33)(25,40,41,47)(26,44,35,43)(28,45,46,42);(1,3,4,7)(2,6,5,8)(10,14)(11,15)(17,20)(18,25)(19,30)(21,37)(22,31)(23,34)(24,40)(26,45)(27,41)(28,43)(29,47)(32,48)(33,39)(35,42)(36,38)(44,46);(10,14)(13,16)(17,21,38,48,22,39,23,30)(18,26,46,47,27,35,28,40)(19,31,37,34,32,20,33,36)(24,41,45,43,29,25,42,44);(2,5)(6,8)(18,27)(20,31)(21,39)(24,29)(28,46)(30,48)(34,36)(42,45)Polycyclic presentation:
2^9|b^2=i,c^2=fi,d^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7515943);