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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1628 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
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)(17,18,22,26)(19,25,31,24)(20,33,32,40)(21,27,37,42)(23,29,34,28)(30,41,47,39)(35,43,36,48)(38,45,46,44);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19)(18,24)(20,34)(21,36)(22,31)(23,32)(25,26)(27,43)(28,40)(29,33)(30,46)(35,37)(38,47)(39,44)(41,45)(42,48);(17,20,34,19,22,32,23,31)(18,25,28,40,26,24,29,33)(21,35,46,30,37,36,38,47)(27,41,44,48,42,39,45,43);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(18,26)(21,37)(24,25)(28,29)(30,47)(33,40)(35,36)(38,46);(3,11)(6,9)(10,16)(14,15)(17,21,23,38,22,37,34,46)(18,27,29,45,26,42,28,44)(19,30,20,35,31,47,32,36)(24,39,40,48,25,41,33,43)Polycyclic presentation:
2^9|a^2=f,c^2=hi,d^2=fi,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, 6716305);