Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480662 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | C2×#535 | C4 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1044 conjugacy classes of subgroups; only the 198 normal subgroups are shown.
This group has 1044 conjugacy classes of subgroups — too many to draw. Only its 198 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:
(1,2,4,8,5,6,3,7)(9,10)(11,21)(12,17)(13,18)(14,28,29,19)(15,20)(16,23,31,38)(22,35)(24,36)(25,34,27,32)(26,39)(30,37,33,40);(2,6)(3,4)(9,11)(10,16)(12,22)(13,26)(14,23)(15,24)(17,30)(18,33)(19,21)(20,31)(25,37)(27,40)(28,36)(29,38)(32,35)(34,39);(1,3,5,4)(2,7,6,8)(9,12,15,13)(10,17,20,18)(11,22,24,26)(14,25,29,27)(16,30,31,33)(19,32,28,34)(21,35,36,39)(23,37,38,40);(2,6)(7,8)(9,13,15,12)(10,18,20,17)(11,22,24,26)(14,27,29,25)(16,30,31,33)(19,34,28,32)(21,35,36,39)(23,37,38,40)Polycyclic presentation:
2^9|a^2=f,c^2=hi,d^2=i,f^2=h,b^a=be,d^a=dh,e^a=eg,g^a=gi,c^b=ch,d^b=di,f^b=fg,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72318204, #72318749, #72318757, … |
| maximal normal subgroup of | #72318204, #72318749, #72318757, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387912);