Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32884 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C4⋊C4 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1342 conjugacy classes of subgroups; only the 330 normal subgroups are shown.
This group has 1342 conjugacy classes of subgroups — too many to draw. Only its 330 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 — 4 generators on 44 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,18,22,19,23,17,31)(15,24,26,20,27,34,32,35)(16,21,30,37,28,36,29,41)(25,38,40,33,39,44,42,43);(13,15,18,26,19,27,17,32)(14,20,22,34,23,35,31,24)(16,25,30,40,28,39,29,42)(21,33,37,44,36,43,41,38);(6,9)(10,12)(13,16)(14,21)(15,25)(17,29)(18,30)(19,28)(20,33)(22,37)(23,36)(24,38)(26,40)(27,39)(31,41)(32,42)(34,44)(35,43);(2,4)(5,7,8,11)(6,10,9,12)(16,28)(21,36)(25,39)(29,30)(33,43)(37,41)(38,44)(40,42)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,e^b=ei,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300654);