Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480684 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1146 conjugacy classes of subgroups; only the 198 normal subgroups are shown.
This group has 1146 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)(3,7)(4,6)(5,8)(9,10)(11,20)(12,24)(13,18)(14,19,27,16)(15,21)(17,25)(22,29)(23,30,35,28)(26,31,37,32)(33,34)(36,38,40,39);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,22)(13,21)(14,23)(16,30)(17,29)(18,20)(19,28)(24,33)(25,34)(26,36)(27,35)(31,39)(32,38)(37,40);(3,8)(5,7)(10,16)(11,21)(12,25)(14,27)(15,28)(17,32)(18,19)(20,30)(24,31)(29,39)(33,38)(36,40);(2,6)(5,7)(9,12,13,25)(10,17,18,24)(11,22,21,34)(14,26,27,37)(15,29,20,33)(16,31,19,32)(23,36,35,40)(28,38,30,39)Polycyclic presentation:
2^9|a^2=g,b^2=ei,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^c=di,f^c=fi,g^c=ghi,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72311237, #72311259, #72311281, … |
| maximal normal subgroup of | #72311237, #72311259, #72311281, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403508);