Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #348098 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2847 |
| 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 | |
1089 conjugacy classes of subgroups; only the 188 normal subgroups are shown.
This group has 1089 conjugacy classes of subgroups — too many to draw. Only its 188 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,8)(3,9,12,16)(4,13,10,7)(6,14,15,11)(17,18)(19,23)(20,24)(21,22)(25,31)(26,28)(27,29)(30,32);(1,3,10,14,5,12,4,11)(2,6,13,16,8,15,7,9)(18,22)(23,28)(24,29)(30,31);(17,19)(18,23)(20,25)(21,26)(22,28)(24,30)(27,32)(29,31);(17,20)(18,24)(19,25)(21,27)(22,29)(23,30)(26,32)(28,31)Polycyclic presentation:
2^9|a^2=h,b^2=ehi,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^c=gi,f^d=fi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72306618, #72306644, #72315844, … |
| maximal normal subgroup of | #72306618, #72306644, #72315844, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 255303);