Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #115179 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C2×#2613 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
2447 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 2447 conjugacy classes of subgroups — too many to draw. Only its 142 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:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,18,20,15)(12,14)(13,17)(16,21)(19,23,24,22);(5,8)(10,14)(12,21)(15,22)(16,17)(18,23)(19,24);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,19)(13,20)(14,22)(16,23)(17,18)(21,24)Polycyclic presentation:
2^9|a^2=g,d^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=dh,d^c=di,g^c=gh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72235262, #72256111, #72260374, … |
| maximal normal subgroup of | #72235262, #72256111, #72260374, … |
| maximal quotient of | C2×#115179, #72260925 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22375);