Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| 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×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1695 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)(3,7)(4,8)(5,6)(17,18)(19,28)(20,32)(21,26)(22,27)(23,36)(24,31)(25,34)(29,44)(30,40)(33,42)(35,39)(37,41)(38,46)(43,47)(45,48);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,29)(21,35)(22,31)(23,30)(25,38)(26,40)(27,28)(32,48)(33,45)(34,47)(36,39)(37,46)(41,43)(42,44);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20)(18,25)(19,29)(21,34)(22,33)(23,37)(24,38)(26,42)(27,41)(28,43)(30,46)(31,45)(32,36)(35,47)(39,48)(40,44);(1,4,5,3)(2,8,6,7)(9,11)(10,14)(12,15)(13,16)(20,33)(25,41)(29,45)(32,42)(34,37)(38,43)(44,48)(46,47);(11,15)(14,16)(17,21,22,23)(18,26,27,36)(19,30,31,35)(20,34,33,37)(24,39,28,40)(25,42,41,32)(29,46,45,47)(38,48,43,44)Polycyclic presentation:
2^9|c^2=fi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ei,d^c=di,g^c=gi,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6445829);