Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #346518 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×H(5,𝔽2) | D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2446 conjugacy classes of subgroups; only the 226 normal subgroups are shown.
This group has 2446 conjugacy classes of subgroups — too many to draw. Only its 226 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 — 4 generators on 24 points:
(5,8)(9,10,15,19)(11,20,12,23)(13,18,14,17)(16,22,24,21);(1,2)(3,5)(4,6)(7,8)(9,11,15,12)(10,16,19,24)(13,22,14,21)(17,23,18,20);(1,3)(2,5)(4,7)(6,8);(9,12,15,11)(10,16,19,24)(13,21,14,22)(17,23,18,20)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,e^d=eh,g^d=gh65 conjugacy classes — view the full character table.
| maximal subgroup of | #72227970 |
| maximal quotient of | #72227967 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 253725);