Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485605 |
| 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 | C2×C22⋊C4 | C24 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1012 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 1012 conjugacy classes of subgroups — too many to draw. Only its 120 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)(6,8)(9,10)(11,16)(12,17)(13,15)(14,18)(19,32)(20,31)(21,27)(22,33)(23,28)(24,34)(25,30)(26,29)(35,38)(36,39)(37,40);(1,2)(3,8)(4,6)(5,7)(10,15)(11,19)(16,30)(17,28)(18,29)(20,22)(21,25)(27,32)(31,39)(33,38)(34,40)(35,36);(9,11,14,19)(10,16,18,32)(12,22,24,20)(13,25,26,21)(15,27,29,30)(17,33,34,31)(23,35,37,36)(28,39,40,38);(2,6)(5,8)(9,12,14,24)(10,17,18,34)(11,20,19,22)(13,23,26,37)(15,28,29,40)(16,31,32,33)(21,35,25,36)(27,38,30,39)Polycyclic presentation:
2^9|c^2=i,d^2=i,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,c^b=ci,d^b=dg,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72318458, #72318739, #72318749 |
| maximal normal subgroup of | #72318458, #72318739, #72318749 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392825);