Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884969 |
| 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 | #2615 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1481 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)(3,4)(5,6)(7,8)(17,18,22,28)(19,29,32,24)(20,33,35,26)(21,27,23,25)(30,40,34,43)(31,41,37,39)(36,44,38,42)(45,48,46,47);(2,6)(4,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,34)(21,37)(22,32)(23,31)(25,39)(26,43)(27,41)(28,29)(30,35)(33,40)(36,45)(38,46)(42,48)(44,47);(1,3)(2,7)(4,6)(5,8)(9,11)(10,13)(12,16)(14,15)(18,25)(20,35)(21,23)(24,39)(26,44)(27,28)(29,41)(30,34)(31,37)(33,42)(40,48)(43,47);(2,6)(3,8)(11,16)(13,15)(17,20,22,35)(18,26,28,33)(19,30,32,34)(21,38,23,36)(24,40,29,43)(25,42,27,44)(31,46,37,45)(39,47,41,48);(1,4,5,7)(2,3,6,8)(10,14)(13,15)(17,21,22,23)(18,27,28,25)(19,31,32,37)(20,36,35,38)(24,39,29,41)(26,42,33,44)(30,45,34,46)(40,48,43,47)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,e^c=ei,g^c=gi,e^d=ei59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6796727);