Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455107 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
861 conjugacy classes of subgroups; only the 202 normal subgroups are shown.
This group has 861 conjugacy classes of subgroups — too many to draw. Only its 202 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) = 256 |
A minimal faithful permutation representation — 4 generators on 44 points:
(2,3)(5,6)(7,9)(8,10)(11,12)(13,14)(15,24)(16,29)(17,21)(18,23)(19,27)(20,32)(22,25)(26,37)(28,35)(30,40)(31,39)(33,38)(34,43)(36,41)(42,44);(1,2,4,3)(13,15,25,41,17,27,18,28)(14,19,22,35,21,24,23,36)(16,30,33,26,31,43,32,42)(20,37,29,34,38,44,39,40);(2,3)(5,7,8,11)(6,9,10,12)(13,16,17,31)(14,20,21,38)(15,26,27,42)(18,33,25,32)(19,34,24,40)(22,39,23,29)(28,43,41,30)(35,37,36,44);(1,3,4,2)(6,10)(9,12)(16,31)(20,38)(26,42)(29,39)(30,43)(32,33)(34,40)(37,44)Polycyclic presentation:
2^9|b^2=gi,c^2=eh,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^c=de,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 382802);