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 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
3412 conjugacy classes of subgroups; only the 196 normal subgroups are shown.
This group has 3412 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) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,6)(3,9)(7,13)(8,10)(11,15)(14,16);(1,2)(3,10)(4,8)(5,7)(6,9)(11,16)(12,14)(13,15);(3,11)(6,13)(9,15)(10,16);(3,9)(6,10)(11,15)(13,16)Polycyclic presentation:
2^9|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 | #72209644, #72209645, #72209860, … |
| maximal normal subgroup of | #72209860, #72209889, #72210342, … |
| maximal quotient of | #72209776, #72209787, #72210483, … |
| holomorph of | C4⋊C4 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 253642);