Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | #2388 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1536 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18,23,28)(19,24,31,39)(20,26,32,42)(21,27,22,25)(29,37,43,48)(30,36,34,38)(33,40,35,41)(44,47,45,46);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,29)(21,34)(22,30)(23,31)(25,36)(26,37)(27,38)(28,39)(32,43)(33,45)(35,44)(40,46)(41,47)(42,48);(1,4,5,2)(3,7,8,6)(9,11)(10,13)(12,16)(14,15)(18,25)(20,32)(21,22)(24,36)(26,41)(27,28)(29,43)(30,34)(37,47)(38,39)(40,42)(46,48);(3,8)(6,7)(11,16)(13,15)(17,20)(18,26)(19,29)(21,35)(22,33)(23,32)(24,37)(25,40)(27,41)(28,42)(30,45)(31,43)(34,44)(36,46)(38,47)(39,48);(10,14)(13,15)(17,21,23,22)(18,27,28,25)(19,30,31,34)(20,33,32,35)(24,36,39,38)(26,40,42,41)(29,44,43,45)(37,47,48,46)Polycyclic presentation:
2^9|a^2=fi,c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,e^c=ei,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, 6903144);