Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485065 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | #534 | C23 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
778 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 778 conjugacy classes of subgroups — too many to draw. Only its 120 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(17,18)(19,24)(20,25)(21,23)(22,26)(27,38)(28,35)(29,39)(30,40)(31,36)(32,41)(33,45)(34,37)(42,46)(43,47)(44,48);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(18,23)(24,35)(25,36)(26,37)(38,46)(39,45)(40,47)(41,48);(17,19,22,29)(18,24,26,39)(20,30,32,27)(21,33,34,28)(23,35,37,45)(25,40,41,38)(31,42,44,43)(36,47,48,46);(2,7)(6,15)(8,12)(9,16)(17,20,22,32)(18,25,26,41)(19,27,29,30)(21,31,34,44)(23,36,37,48)(24,38,39,40)(28,42,33,43)(35,46,45,47)Polycyclic presentation:
2^9|b^2=g,c^2=i,d^2=i,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^e=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392301);