Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485611 |
| 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×C22⋊C4 | C24 |
| prime radical | #23445 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1045 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 1045 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)(11,19)(12,16)(13,17)(14,18)(15,21)(20,33)(22,34)(23,37)(24,30)(25,31)(26,32)(27,35)(28,36)(29,38)(39,40);(1,2)(3,8)(4,6)(5,7)(9,11,14,22)(10,15,18,28)(12,23,24,20)(13,21,26,36)(16,29,30,27)(17,19,32,34)(25,38,39,35)(31,37,40,33);(12,24)(13,26)(16,30)(20,23)(21,36)(27,29)(31,40)(33,37);(2,6)(5,8)(9,12,14,24)(10,16,18,30)(11,20,22,23)(13,25,26,39)(15,27,28,29)(17,31,32,40)(19,33,34,37)(21,35,36,38)Polycyclic presentation:
2^9|b^2=i,d^2=i,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,d^b=dgi,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 | #72283562, #72283702, #72283765, … |
| maximal normal subgroup of | #72283562, #72283702, #72283765, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392811);