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×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1398 conjugacy classes of subgroups; only the 274 normal subgroups are shown.
This group has 1398 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)(7,10,14,17)(8,11)(9,12)(13,16,19,20)(15,18)(21,22,24,28,25,27,31,34)(23,26,29,33,30,32,35,36);(5,7,9,14)(6,10)(8,13,15,19)(11,16)(12,17)(18,20)(21,23,24,29,25,30,31,35)(22,26,28,33,27,32,34,36);(2,4)(8,15)(11,18)(13,19)(16,20);(5,8,9,15)(6,11,12,18)(7,13,14,19)(10,16,17,20)(22,27)(26,32)(28,34)(33,36)Polycyclic presentation:
2^9|a^2=g,b^2=egh,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, 103161);