Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #487707 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C4⋊C4 | C24 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
876 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 876 conjugacy classes of subgroups — too many to draw. Only its 118 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:
(2,6)(3,4)(7,8)(9,10)(11,19)(12,16)(13,26)(14,18)(15,22)(17,28)(20,35)(21,34)(23,31)(24,38)(25,40)(27,30)(29,36)(32,39)(33,37);(1,2)(3,6)(4,8)(5,7)(12,23)(16,31)(17,26)(19,34)(20,24)(25,37)(29,32)(36,39);(3,4)(6,8)(9,11,14,21)(10,15,18,30)(12,24,23,20)(13,27,28,22)(16,32,31,29)(17,34,26,19)(25,36,37,39)(33,35,40,38);(1,3,5,4)(2,6,7,8)(9,12,14,23)(10,16,18,31)(11,20,21,24)(13,25,28,37)(15,29,30,32)(17,33,26,40)(19,35,34,38)(22,36,27,39)Polycyclic presentation:
2^9|c^2=i,d^2=gi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,h^a=hi,d^b=di,e^b=eg,f^b=fhi,d^c=dgi,e^c=eh,f^c=fi,f^e=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 395149);