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 | |
1572 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)(19,24)(20,21)(22,25)(23,27)(26,36)(28,32)(29,33)(30,39)(31,40)(34,35)(37,41)(38,43)(42,48)(44,46)(45,47);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19)(18,24)(20,33)(21,29)(22,37)(23,32)(25,41)(26,40)(27,28)(30,35)(31,36)(34,39)(38,45)(42,44)(43,47)(46,48);(17,20,21,27,23,26,36,18)(19,28,31,33,32,24,29,40)(22,34,35,43,38,42,48,25)(30,44,37,47,46,39,45,41);(2,7)(6,9)(8,15)(14,16)(17,21,23,36)(18,20,27,26)(19,29,32,31)(22,35,38,48)(24,33,28,40)(25,34,43,42)(30,45,46,37)(39,47,44,41);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,22,36,48,23,38,21,35)(18,25,26,42,27,43,20,34)(19,30,29,45,32,46,31,37)(24,39,33,47,28,44,40,41)Polycyclic presentation:
2^9|b^2=f,c^2=h,d^2=i,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, 6716866);