Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511775 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | #23908 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1100 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 1100 conjugacy classes of subgroups — too many to draw. Only its 140 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:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,17)(14,19)(18,26)(20,27)(21,28)(22,30)(23,32)(24,33)(25,31)(29,37)(34,39)(35,40)(36,38);(2,5)(6,8)(9,11,14,22)(10,15,19,30)(12,20,23,34)(13,21,26,37)(16,27,32,39)(17,28,18,29)(24,35,25,36)(31,38,33,40);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,20)(13,25)(14,23)(15,27)(17,33)(18,31)(19,32)(21,36)(22,34)(24,26)(28,40)(29,38)(30,39)(35,37);(10,17)(12,23)(13,26)(15,28)(16,31)(18,19)(20,34)(21,37)(27,38)(29,30)(32,33)(39,40)Polycyclic presentation:
2^9|b^2=i,c^2=e,g^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gi,h^a=hi,d^c=di,g^c=ghi,f^d=fh,g^d=gi,g^f=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72281983, #72282650, #72285468, … |
| maximal normal subgroup of | #72281983, #72282650, #72285468, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419317);