Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #397903 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2856 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14910 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
859 conjugacy classes of subgroups; only the 204 normal subgroups are shown.
This group has 859 conjugacy classes of subgroups — too many to draw. Only its 204 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) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12,8,14,6,13)(3,15,19,22,20,29,18,9)(4,17,23,31,25,30,24,10)(5,16,27,21,28,32,26,11);(1,3,6,18,8,20,7,19)(2,9,12,15,14,22,13,29)(4,21,23,32,25,11,24,16)(5,17,26,10,28,30,27,31);(1,4,6,24,8,25,7,23)(2,10,12,17,14,31,13,30)(3,16,19,21,20,32,18,11)(5,22,26,15,28,9,27,29);(1,5,7,27,8,28,6,26)(2,11,12,16,14,21,13,32)(3,17,19,31,20,30,18,10)(4,22,23,29,25,9,24,15)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=f,d^2=g,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,f^b=fi,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312266);