Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470884 |
| 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 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1375 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)(11,13)(12,14)(15,16)(17,18)(19,24)(20,31)(21,26)(22,36)(23,27)(25,33)(28,37)(29,43)(30,42)(32,45)(34,46)(35,39)(38,44)(40,47)(41,48);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,32)(21,23)(22,30)(25,40)(26,27)(28,39)(29,34)(31,45)(33,47)(35,37)(36,42)(38,41)(43,46)(44,48);(1,3,5,4)(2,7,6,8)(11,15)(13,16)(17,20,22,33,23,34,37,48)(18,25,28,31,27,41,36,46)(19,29,30,44,21,32,35,47)(24,38,39,43,26,40,42,45);(11,15)(13,16)(17,21,23,19)(18,26,27,24)(20,29,34,32)(22,35,37,30)(25,38,41,40)(28,42,36,39)(31,43,46,45)(33,44,48,47);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(18,27)(19,21)(25,41)(28,36)(29,32)(30,35)(31,46)(44,47)Polycyclic presentation:
2^9|c^2=fg,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6382173);