Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| 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 | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1725 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,8)(5,6)(17,18)(19,24)(20,32)(21,26)(22,35)(23,27)(25,34)(28,42)(29,38)(30,39)(31,45)(33,41)(36,40)(37,43)(44,47)(46,48);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,28)(21,30)(22,31)(23,29)(25,37)(26,39)(27,38)(32,42)(33,44)(34,43)(35,45)(36,46)(40,48)(41,47);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20)(18,25)(19,28)(21,34)(22,36)(23,33)(24,37)(26,41)(27,40)(29,44)(30,43)(31,46)(32,35)(38,48)(39,47)(42,45);(1,4,5,3)(2,8,6,7)(9,11)(10,14)(12,15)(13,16)(19,29)(20,33)(24,38)(25,40)(30,31)(32,41)(34,36)(39,45);(11,15)(14,16)(17,21,23,22)(18,26,27,35)(19,30,29,31)(20,34,33,36)(24,39,38,45)(25,41,40,32)(28,43,44,46)(37,47,48,42)Polycyclic presentation:
2^9|c^2=f,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,d^c=di,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6446015);