Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499823 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1595 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)(17,18)(19,25)(20,34)(21,27)(22,28)(23,38)(24,29)(26,36)(30,40)(31,41)(32,47)(33,42)(35,48)(37,44)(39,45)(43,46);(2,6)(7,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,25)(20,35)(21,30)(22,31)(23,32)(24,33)(26,43)(27,40)(28,41)(29,42)(34,48)(36,46)(37,45)(38,47)(39,44);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,16)(14,15)(17,20)(18,26)(19,30)(21,24)(22,36)(23,37)(25,39)(27,28)(29,44)(31,45)(32,46)(33,35)(34,38)(40,47)(41,48)(42,43);(1,4,5,3)(2,8,6,7)(11,16)(13,15)(17,21,24,20)(18,27,29,34)(19,30,33,35)(22,36,23,37)(25,40,42,48)(26,38,44,28)(31,46,32,45)(39,41,43,47);(10,14)(13,15)(17,22,24,23)(18,28,29,38)(19,31,33,32)(20,36,21,37)(25,41,42,47)(26,27,44,34)(30,45,35,46)(39,48,43,40)Polycyclic presentation:
2^9|c^2=f,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6408774);