Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6885065 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1551 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,8,4,6)(9,10,12,14)(11,15,16,13)(17,18,23,36,22,27,21,26)(19,24,37,48,29,39,30,40)(20,31,35,25,33,43,34,42)(28,44,46,38,32,47,45,41);(1,2)(3,6)(4,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,19)(18,24)(20,32)(21,30)(22,29)(23,37)(25,41)(26,40)(27,39)(28,33)(31,47)(34,46)(35,45)(36,48)(38,42)(43,44);(1,2)(3,8)(4,6)(5,7)(11,16)(13,15)(17,20,22,33)(18,25,27,42)(19,28,29,32)(21,35,23,34)(24,38,39,41)(26,43,36,31)(30,46,37,45)(40,47,48,44);(1,3,5,4)(2,6,7,8)(10,14)(13,15)(19,29)(20,33)(24,39)(25,42)(30,37)(31,43)(34,35)(40,48);(1,4,5,3)(2,6,7,8)(19,29)(24,39)(28,32)(30,37)(38,41)(40,48)(44,47)(45,46)Polycyclic presentation:
2^9|a^2=gh,c^2=i,d^2=fhi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6896266);