Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682360 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1336 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)(17,18,22,26)(19,27,30,23)(20,31,34,38)(21,35,36,25)(24,32,40,28)(29,37,43,41)(33,46,47,44)(39,42,48,45);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,22,30)(18,23,26,27)(20,32,34,28)(21,29,36,43)(24,38,40,31)(25,37,35,41)(33,45,47,42)(39,46,48,44);(2,6)(5,7)(10,14)(15,16)(17,20,19,28,22,34,30,32)(18,24,23,31,26,40,27,38)(21,33,29,42,36,47,43,45)(25,39,37,44,35,48,41,46);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,21,22,36)(18,25,26,35)(19,29,30,43)(20,33,34,47)(23,37,27,41)(24,39,40,48)(28,42,32,45)(31,44,38,46);(3,8)(5,7)(9,12,13,11)(10,15,14,16)(20,34)(21,36)(24,40)(25,35)(28,32)(29,43)(31,38)(37,41)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgi,d^2=h,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6784386);