Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #457592 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2842 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
738 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 738 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(3,7)(9,10,13,17,14,18,24,34)(11,15,22,30,21,31,20,29)(12,23,25,16,26,39,27,33)(19,35,38,28,37,40,36,32);(1,2,4,5)(3,6,7,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,32)(17,29)(18,31)(22,24)(23,35)(25,36)(26,37)(27,38)(28,33)(30,34)(39,40);(1,3)(2,6)(4,7)(5,8)(9,12,24,25,14,26,13,27)(10,16,17,23,18,33,34,39)(11,19,20,38,21,37,22,36)(15,28,30,35,31,32,29,40);(3,7)(6,8)(9,13,14,24)(10,17,18,34)(11,20,21,22)(12,25,26,27)(15,29,31,30)(16,23,33,39)(19,36,37,38)(28,40,32,35)Polycyclic presentation:
2^9|a^2=efg,b^2=egh,c^2=ef,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,d^c=de,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 381384);