Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #195963 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1300 conjugacy classes of subgroups; only the 274 normal subgroups are shown.
This group has 1300 conjugacy classes of subgroups — too many to draw. Only its 274 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)(5,6,8,12,9,11,15,18)(7,10,13,17,14,16,19,20)(21,22,26,29,27,30,25,23)(24,28,32,36,33,35,34,31);(5,7,8,13,9,14,15,19)(6,10,12,17,11,16,18,20)(21,23,26,30,27,29,25,22)(24,31,32,35,33,36,34,28);(2,4)(21,24,27,33)(22,28,30,35)(23,31,29,36)(25,34,26,32);(6,11)(10,16)(12,18)(17,20)(21,25,27,26)(22,29,30,23)(24,32,33,34)(28,31,35,36)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=h,d^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=eh,g^b=gh,d^c=dh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 103162);