Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #488895 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | C23 | #447 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
651 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10)(17,18,24,27)(19,31,26,35)(20,32,25,33)(21,37,22,42)(23,46,40,30)(28,45,29,41)(34,39,47,44)(36,43,48,38);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,19,24,26)(18,25,27,20)(21,38,22,43)(23,47,40,34)(28,44,29,39)(30,36,46,48)(31,42,35,37)(32,45,33,41);(17,20,24,25)(18,26,27,19)(21,39,22,44)(23,48,40,36)(28,43,29,38)(30,34,46,47)(31,41,35,45)(32,37,33,42);(18,27)(19,26)(28,29)(30,46)(32,33)(34,47)(37,42)(38,43)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=i,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fh,g^a=gi,h^a=hi,d^b=di,e^b=eg,f^b=fh,g^b=gi,h^b=hi,e^c=eh,f^c=fh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 396334);