Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667964 |
| 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 | #60576 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1623 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)(17,18,23,29)(19,30,32,25)(20,33,36,44)(21,38,24,27)(22,28,40,48)(26,35,46,37)(31,43,41,45)(34,47,39,42);(1,3,5,11)(2,6,8,9)(4,10,13,16)(7,14,12,15)(17,19,23,32)(18,25,29,30)(20,34,36,39)(21,31,24,41)(22,37,40,35)(26,28,46,48)(27,43,38,45)(33,42,44,47);(17,20,23,36)(18,26,29,46)(19,22,32,40)(21,35,24,37)(25,42,30,47)(27,44,38,33)(28,45,48,43)(31,39,41,34);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(17,21,23,24)(18,27,29,38)(19,31,32,41)(20,35,36,37)(22,39,40,34)(25,43,30,45)(26,44,46,33)(28,47,48,42);(3,11)(6,9)(10,16)(14,15)(17,22,23,40)(18,28,29,48)(19,20,32,36)(21,34,24,39)(25,44,30,33)(26,45,46,43)(27,42,38,47)(31,37,41,35)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=i,d^2=f,e^2=i,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,h^c=hi,e^d=ei,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6579829);