Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6807031 |
| 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 | #3060 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1585 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,9,11,6)(4,12,13,7)(10,14,16,15)(17,18)(19,25)(20,22)(21,26)(23,27)(24,28)(29,39)(30,32)(31,41)(33,36)(34,42)(35,43)(37,44)(38,40)(45,46)(47,48);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19,24,31)(18,25,28,41)(20,32,27,40)(21,34,35,47)(22,30,23,38)(26,42,43,48)(29,36,45,37)(33,46,44,39);(17,20,22,28,24,27,23,18)(19,25,38,40,31,41,30,32)(21,33,36,43,35,44,37,26)(29,39,47,48,45,46,34,42);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(21,35)(26,43)(29,45)(33,44)(34,47)(36,37)(39,46)(42,48);(3,11)(6,9)(10,16)(14,15)(17,21,23,37,24,35,22,36)(18,26,27,44,28,43,20,33)(19,29,30,34,31,45,38,47)(25,39,32,42,41,46,40,48)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=hi,d^2=f,e^2=h,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=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6716311);