Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1466 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)(11,15)(12,16)(13,14)(17,18,21,23)(19,27,29,42)(20,25,31,37)(22,35,34,26)(24,38,36,30)(28,43,32,46)(33,48,44,41)(39,47,40,45);(1,3)(2,6)(4,7)(5,8)(10,14)(11,12)(18,23)(20,31)(24,36)(26,35)(27,42)(28,32)(33,44)(45,47);(9,10)(11,16)(12,15)(13,14)(17,19,22,30,21,29,34,38)(18,24,26,27,23,36,35,42)(20,32,33,47,31,28,44,45)(25,40,41,46,37,39,48,43);(1,2,5,4)(3,7,8,6)(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,28)(21,31)(22,33)(23,37)(24,39)(26,41)(27,43)(29,32)(30,45)(34,44)(35,48)(36,40)(38,47)(42,46);(1,4,5,2)(3,7,8,6)(9,12,13,11)(10,15,14,16)(19,29)(20,31)(24,36)(25,37)(27,42)(30,38)(33,44)(41,48)Polycyclic presentation:
2^9|a^2=fh,c^2=g,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6957850);