Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476106 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
958 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 958 conjugacy classes of subgroups — too many to draw. Only its 162 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:
(1,2,4,5)(3,7,8,6)(9,10)(11,19)(12,16)(13,27)(14,18)(15,22)(17,28)(20,36)(21,31)(23,30)(24,34)(25,33)(26,35)(29,39)(32,37)(38,40);(2,5)(6,7)(9,11)(10,15)(12,20)(13,21)(14,23)(16,32)(17,30)(18,31)(19,27)(22,28)(24,37)(25,39)(26,38)(29,35)(33,36)(34,40);(1,3,4,8)(2,6,5,7)(9,12)(10,16)(11,20)(13,24)(14,26)(15,29)(17,33)(18,35)(19,36)(21,37)(22,39)(23,38)(25,28)(27,34)(30,40)(31,32);(3,8)(6,7)(11,21)(12,24)(13,28)(15,30)(16,33)(17,27)(19,31)(22,23)(25,26)(34,35)(36,40)(37,39)Polycyclic presentation:
2^9|a^2=f,c^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=def,e^b=ei,d^c=de,e^c=ei,h^c=hi,e^d=ei50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 384443);