Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #346446 |
| 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 | #62817 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
3046 conjugacy classes of subgroups; only the 196 normal subgroups are shown.
This group has 3046 conjugacy classes of subgroups — too many to draw. Only its 196 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 — 4 generators on 24 points:
(5,8)(9,10,13,16)(11,17,19,24)(12,20,22,15)(14,21,23,18);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,21)(13,19)(15,24)(16,23)(17,20)(18,22);(1,3)(2,5)(4,7)(6,8);(12,22)(15,20)(17,24)(18,21)Polycyclic presentation:
2^9|a^2=h,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,e^d=eh59 conjugacy classes — view the full character table.
| maximal subgroup of | #72209762, #72209835, #72209889, … |
| maximal normal subgroup of | #72209835, #72209889, #72213838, … |
| maximal quotient of | #72209757, #72219246, #72227597, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 253652);