Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7200022 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23854 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1652 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,7,4,6)(11,15)(14,16)(17,18)(19,30)(20,26)(21,29)(22,28)(23,27)(24,34)(25,35)(31,40)(32,46)(33,41)(36,44)(37,45)(38,43)(39,47)(42,48);(1,2)(3,6)(4,7)(5,8)(9,10)(11,16)(12,13)(14,15)(17,19)(18,24)(20,32)(21,37)(22,34)(23,33)(25,42)(26,40)(27,45)(28,30)(29,41)(31,36)(35,47)(38,48)(39,43)(44,46);(1,3,5,4)(2,6,8,7)(10,13)(14,16)(17,20)(18,25)(19,31)(21,35)(22,36)(23,38)(24,39)(26,27)(28,43)(29,44)(30,42)(32,34)(33,47)(37,48)(40,41)(45,46);(1,3,5,4)(2,7,8,6)(9,11,12,15)(10,14,13,16)(17,20)(18,26)(19,32)(21,38)(22,36)(23,35)(24,40)(25,27)(28,44)(29,43)(30,46)(31,34)(33,47)(37,48)(39,41)(42,45);(1,4,5,3)(2,7,8,6)(10,13)(14,16)(17,21,22,23)(18,27,28,29)(19,33,34,37)(20,35,36,38)(24,41,30,45)(25,26,43,44)(31,47,32,48)(39,40,42,46)Polycyclic presentation:
2^9|a^2=fi,c^2=fi,d^2=fh,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7131386);