Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478213 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2842 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C22×Q8 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
802 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 802 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,11,15,12,17,10,16)(7,18,22,31,23,33,21,13)(8,9,25,28,26,29,30,14)(19,27,36,34,35,20,24,32);(5,7,10,21,12,23,11,22)(6,13,15,18,17,31,16,33)(8,24,25,19,26,36,30,35)(9,27,14,32,29,20,28,34);(5,8,10,25,12,26,11,30)(6,14,15,29,17,28,16,9)(7,19,21,36,23,35,22,24)(13,20,18,34,31,27,33,32);(2,4)(5,9,10,28,12,29,11,14)(6,8,16,25,17,26,15,30)(7,20,22,34,23,27,21,32)(13,24,18,35,31,36,33,19)Polycyclic presentation:
2^9|a^2=efgi,b^2=e,c^2=e,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=def,f^b=fi,g^b=gi,d^c=de,g^c=gi,h^c=hi,e^d=ei,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386476);