Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #487659 |
| 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 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1011 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 1011 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(3,4)(7,8)(9,10)(11,15)(12,14)(13,16)(17,24)(18,21)(19,23)(20,22);(1,2,5,7)(3,6,4,8)(10,14)(11,17)(15,23)(16,22)(18,19)(21,24);(3,4)(6,8)(9,11)(10,15)(12,19)(13,17)(14,21)(16,24)(18,20)(22,23);(1,3,5,4)(2,6,7,8)(11,17)(15,24)(18,19)(21,23)Polycyclic presentation:
2^9|b^2=g,d^2=g,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,c^b=ci,e^b=eg,f^b=fhi,d^c=dgi,e^c=eh,f^e=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 395249);