Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10557 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #394 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1442 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
668 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 668 conjugacy classes of subgroups — too many to draw. Only its 126 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 — 3 generators on 40 points:
(1,2)(3,4,6,8)(5,7)(9,10,14,18,15,20,13,19)(11,21,24,32,25,35,23,16)(12,26,29,34,30,36,28,17)(22,38,39,33,40,31,27,37);(1,3)(2,4,7,8)(5,6)(9,11,13,23,15,25,14,24)(10,16,18,21,20,32,19,35)(12,27,28,40,30,39,29,22)(17,33,26,31,34,37,36,38);(1,4)(2,6,7,3)(5,8)(9,12,13,28,15,30,14,29)(10,17,18,26,20,34,19,36)(11,22,23,27,25,40,24,39)(16,31,21,37,32,38,35,33)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,d^2=hi,e^2=hi,f^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=di,f^b=fh,g^b=gi,e^c=ei,f^c=fh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 27846);