Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #161262 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2468 conjugacy classes of subgroups; only the 338 normal subgroups are shown.
This group has 2468 conjugacy classes of subgroups — too many to draw. Only its 338 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)(7,13,15,10)(8,16)(9,12)(11,17)(14,18,20,19)(21,22)(23,29)(24,32)(25,27)(26,30)(28,34)(31,36)(33,35);(5,7,9,15)(6,10,12,13)(8,14,17,20)(11,18,16,19)(21,23,25,31)(22,26,27,35)(24,33,34,30)(28,36,32,29);(2,4)(8,17)(11,16)(14,20)(18,19);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(22,27)(26,35)(28,32)(29,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=hi,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,e^b=ei,g^b=ghi,d^c=dhi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92293);