Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7309060 |
| 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 | #42908 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1596 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)(17,18,24,39,23,29,22,28)(19,30,32,43,33,25,34,42)(20,27,36,26,21,38,37,45)(31,41,48,44,35,46,47,40);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,25)(20,35)(21,31)(22,34)(23,33)(24,32)(26,44)(27,41)(28,43)(29,30)(36,47)(37,48)(38,46)(39,42)(40,45);(9,11)(10,13)(12,16)(14,15)(17,20)(18,26)(19,31)(21,23)(22,36)(24,37)(25,40)(27,39)(28,38)(29,45)(30,44)(32,47)(33,35)(34,48)(41,43)(42,46);(3,8)(5,7)(10,14)(11,16)(17,21,23,20)(18,27,29,38)(19,31,33,35)(22,36,24,37)(25,41,30,46)(26,28,45,39)(32,48,34,47)(40,42,44,43);(2,6)(5,7)(10,14)(13,15)(17,22,23,24)(18,28,29,39)(19,32,33,34)(20,36,21,37)(25,42,30,43)(26,38,45,27)(31,47,35,48)(40,46,44,41)Polycyclic presentation:
2^9|a^2=gi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7216331);