Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129781 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×#117 | C23 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#520 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
797 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 797 conjugacy classes of subgroups — too many to draw. Only its 108 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) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5,6,7)(9,10)(11,15,12,17)(13,18)(14,19)(16,20,23,21)(22,24);(1,2)(3,7)(4,6)(5,8)(10,15,19,17)(16,18,23,24);(9,11,14,12)(10,16,19,23)(13,21,22,20)(15,18,17,24)Polycyclic presentation:
2^9|a^2=f,b^2=gi,c^2=i,d^2=hi,e^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,f^b=fghi,e^c=ei,f^c=fi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72227828 |
| maximal normal subgroup of | #72227828 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37024);