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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1416 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(11,15)(14,16)(17,18,21,34,22,24,23,27)(19,28,36,25,31,40,30,37)(20,26,35,41,33,38,32,42)(29,43,44,39,45,48,46,47);(1,3)(2,5)(4,8)(6,7)(10,13)(11,15)(18,24)(25,37)(26,38)(27,34)(28,40)(39,47)(41,42)(43,48);(9,10)(11,14)(12,13)(15,16)(17,19,22,31)(18,25,24,37)(20,29,33,45)(21,30,23,36)(26,39,38,47)(27,40,34,28)(32,44,35,46)(41,43,42,48);(3,8)(5,7)(10,13)(14,16)(17,20)(18,26)(19,29)(21,35)(22,33)(23,32)(24,38)(25,39)(27,42)(28,43)(30,46)(31,45)(34,41)(36,44)(37,47)(40,48);(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,27,24,34)(19,30,31,36)(20,32,33,35)(25,40,37,28)(26,41,38,42)(29,44,45,46)(39,43,47,48)Polycyclic presentation:
2^9|a^2=fg,c^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,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, 7055976);