Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #184767 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator, prime radical | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1418 conjugacy classes of subgroups; only the 322 normal subgroups are shown.
This group has 1418 conjugacy classes of subgroups — too many to draw. Only its 322 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:
(5,6)(7,10)(8,9)(11,12)(13,14,17,22,18,21,32,36)(15,23,26,37,25,19,30,34)(16,27,28,38,29,20,31,35)(24,33,41,43,40,39,42,44);(1,2)(3,4)(6,9)(10,12)(13,15,18,25)(14,19,21,23)(16,24,29,40)(17,30,32,26)(20,33,27,39)(22,37,36,34)(28,42,31,41)(35,44,38,43);(2,4)(13,16,18,29)(14,20,21,27)(15,24,25,40)(17,31,32,28)(19,33,23,39)(22,38,36,35)(26,42,30,41)(34,44,37,43);(5,7,8,11)(6,10,9,12)(15,25)(19,23)(24,40)(26,30)(33,39)(34,37)(41,42)(43,44)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=i,d^2=ef,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,d^b=di,h^b=hi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265871);