Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #128057 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#342 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C23×Q8 | C23 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ C24×Q8 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1 | |
1396 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 1396 conjugacy classes of subgroups — too many to draw. Only its 154 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 — 3 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,17,14,18,21,24)(11,15,19,22,12,16,20,23);(1,3)(2,5,6,7)(4,8)(10,15,18,16)(17,23,24,22);(9,11,14,12)(10,15,18,16)(13,20,21,19)(17,23,24,22)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=i,d^2=i,e^2=i,b^a=bd,c^a=cg,d^a=dg,f^a=fhi,g^a=gi,d^b=dh,e^b=egi,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72227407, #72227408, #72227418, … |
| maximal normal subgroup of | #72227407, #72227408, #72227418, … |
| maximal quotient of | C2×#128060, #72267712, #72267713, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35256);