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 | #39335 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1787 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)(3,6)(4,5)(7,8)(17,18,23,27)(19,24,31,39)(20,32,34,42)(21,28,22,26)(25,36,40,33)(29,43,45,47)(30,41,35,38)(37,44,48,46);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,29)(21,35)(22,30)(23,31)(25,37)(26,41)(27,39)(28,38)(32,43)(33,46)(34,45)(36,44)(40,48)(42,47);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20,23,34)(18,25,27,40)(19,29,31,45)(21,33,22,36)(24,37,39,48)(26,32,28,42)(30,44,35,46)(38,47,41,43);(9,11)(10,14)(12,15)(13,16)(17,21,23,22)(18,26,27,28)(19,30,31,35)(20,33,34,36)(24,38,39,41)(25,32,40,42)(29,44,45,46)(37,47,48,43);(11,15)(14,16)(18,27)(19,31)(25,40)(26,28)(29,45)(30,35)(32,42)(44,46)Polycyclic presentation:
2^9|a^2=i,c^2=fi,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=ei,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, 6445430);