Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150138 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#357 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | C24 | SD32 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
624 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 624 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 68 points:
(1,2)(3,4)(5,6,9,15,12,17,11,35)(7,18,20,49,23,51,22,58)(8,24,27,59,30,61,29,66)(10,31,26,47,34,42,33,16)(13,36,37,19,39,53,45,55)(14,40,43,65,44,64,60,28)(21,56,52,41,57,68,32,50)(25,46,62,67,54,48,63,38);(5,7,12,23)(6,13,17,39)(8,25,30,54)(9,22,11,20)(10,32,34,52)(14,41,44,50)(15,45,35,37)(16,46,47,48)(18,24,51,61)(19,28,55,65)(21,26,57,33)(27,63,29,62)(31,38,42,67)(36,40,53,64)(43,56,60,68)(49,66,58,59);(2,4)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=ghi,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dgi,f^b=fi,g^b=gh,h^b=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59359);