Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | #35687 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2224 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)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,29)(19,25,34,43)(20,35,33,26)(21,27,24,30)(22,39,31,28)(32,42,41,48)(36,46,38,44)(37,47,40,45);(1,3)(2,5)(4,8)(6,7)(9,11,12,15)(10,13,14,16)(17,19)(18,25)(20,36)(21,32)(22,40)(23,34)(24,41)(26,44)(27,42)(28,47)(29,43)(30,48)(31,37)(33,38)(35,46)(39,45);(10,14)(13,16)(17,20,23,33)(18,26,29,35)(19,31,34,22)(21,37,24,40)(25,39,43,28)(27,45,30,47)(32,38,41,36)(42,46,48,44);(2,6)(5,7)(17,21,23,24)(18,27,29,30)(19,32,34,41)(20,37,33,40)(22,36,31,38)(25,42,43,48)(26,45,35,47)(28,44,39,46);(3,8)(5,7)(17,22,23,31)(18,28,29,39)(19,33,34,20)(21,38,24,36)(25,35,43,26)(27,46,30,44)(32,37,41,40)(42,45,48,47)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=i,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6585215);