Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806815 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #61262 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1589 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,9)(4,8)(5,7)(6,11)(10,16)(12,14)(13,15)(17,18)(19,29)(20,32)(21,27)(22,26)(23,39)(24,28)(25,31)(30,45)(33,43)(34,41)(35,47)(36,42)(37,48)(38,44)(40,46);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19,22,31)(18,25,26,29)(20,24,34,23)(21,35,36,33)(27,43,42,47)(28,32,39,41)(30,38,46,37)(40,44,45,48);(17,20,24,31,22,34,23,19)(18,25,28,41,26,29,39,32)(21,33,38,46,36,35,37,30)(27,40,44,47,42,45,48,43);(2,7)(6,9)(8,15)(14,16)(18,26)(25,29)(27,42)(28,39)(32,41)(40,45)(43,47)(44,48);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,21,24,38,22,36,23,37)(18,27,28,44,26,42,39,48)(19,30,20,33,31,46,34,35)(25,40,41,47,29,45,32,43)Polycyclic presentation:
2^9|b^2=f,c^2=h,e^2=fhi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6716862);