Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129741 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C26 | D4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1296 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 1296 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5,6,7)(9,10)(11,17)(12,20,22,15)(13,16)(14,18,23,21)(19,24);(1,2)(3,7)(4,6)(5,8)(9,11)(10,14,16,23)(12,21)(13,19)(15,24,20,17)(18,22);(12,22)(15,20)(17,24)(18,21)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=hi,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=ei,f^b=fghi,d^c=di,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72227596, #72227893, #72227898, … |
| maximal normal subgroup of | #72227596, #72227893, #72227898, … |
| maximal quotient of | #72227591 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37001);