Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12093 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C24 |
| socle | C22 | C2×B(4,𝔽2) |
| prime radical | #23443 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1046 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 1046 conjugacy classes of subgroups — too many to draw. Only its 138 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)(15,22)(20,31)(21,32)(23,28)(24,29)(25,30)(26,34)(27,35)(33,39)(36,38)(37,40);(1,2,4,6)(3,8,7,5)(9,11)(10,15)(12,21)(13,22)(14,20)(16,27)(17,19)(18,26)(23,35)(24,33)(25,34)(28,32)(29,37)(30,31)(36,40)(38,39);(11,20)(13,25)(15,26)(19,31)(21,33)(23,36)(27,37)(32,39);(9,12,14,24)(10,16,18,29)(11,21,20,33)(13,23,25,36)(15,27,26,37)(17,28,30,38)(19,32,31,39)(22,35,34,40)Polycyclic presentation:
2^9|b^2=g,d^2=i,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,c^b=ci,e^b=eg,f^b=fhi,e^c=eh,f^e=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391476);