Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501090 |
| 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, prime radical, cosocle | C22×D4 | C24 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
910 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 910 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,17)(11,19,21,15)(12,16,24,32)(14,18,26,34)(20,29,38,37)(22,35,27,30)(23,40,39,31)(25,36,28,33);(1,3,5,4)(2,6,8,7)(9,11,13,21)(10,15,17,19)(12,23,24,39)(14,27,26,22)(16,31,32,40)(18,35,34,30)(20,28,38,25)(29,36,37,33);(1,3,5,4)(2,7,8,6)(9,12,13,24)(10,16,17,32)(11,20)(14,28)(15,29)(18,36)(19,37)(21,38)(22,39,27,23)(25,26)(30,40,35,31)(33,34);(1,4,5,3)(2,7,8,6)Polycyclic presentation:
2^9|a^2=i,b^2=e,c^2=ef,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=dei,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,g^c=ghi,h^c=hi,g^f=gi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410660);