Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #35660 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2235 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)(9,10,12,14)(11,13,15,16)(17,18,22,24)(19,28,29,40)(20,32,34,26)(21,27,23,36)(25,37,38,31)(30,45,44,43)(33,42,35,41)(39,47,48,46);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(18,24)(19,29)(26,32)(27,36)(30,44)(31,37)(41,42)(46,47);(11,15)(13,16)(17,19,22,29)(18,25,24,38)(20,30,34,44)(21,31,23,37)(26,39,32,48)(27,40,36,28)(33,46,35,47)(41,43,42,45);(2,6)(5,7)(17,20)(18,26)(19,30)(21,35)(22,34)(23,33)(24,32)(25,39)(27,42)(28,43)(29,44)(31,47)(36,41)(37,46)(38,48)(40,45);(3,8)(5,7)(11,15)(13,16)(17,21,22,23)(18,27,24,36)(19,31,29,37)(20,33,34,35)(25,40,38,28)(26,41,32,42)(30,46,44,47)(39,43,48,45)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6652909);