Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #97058 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C25 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | #35088 | C2 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1821 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 1821 conjugacy classes of subgroups — too many to draw. Only its 154 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 44 points:
(5,6)(7,11,12,9)(8,10)(13,14,17,21,18,22,28,34)(15,19,25,31,24,30,38,42)(16,20,27,33,26,32,39,43)(23,35,37,44,36,29,40,41);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,19)(16,23)(17,25)(18,24)(20,29)(21,31)(22,30)(26,36)(27,37)(28,38)(32,35)(33,41)(34,42)(39,40)(43,44);(2,4)(13,16)(14,20)(15,23)(17,27)(18,26)(19,29)(21,33)(22,32)(24,36)(25,37)(28,39)(30,35)(31,41)(34,43)(38,40)(42,44)Polycyclic presentation:
2^9|a^2=g,g^2=i,b^a=bd,c^a=ce,d^a=dh,c^b=cf,e^b=ei,g^b=gh,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22548);