Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C22×#102 |
| prime radical | C22×#135 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
969 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,3,6,4,5,7,8)(9,10)(11,21)(12,17)(13,26)(14,18)(15,20)(16,23)(19,28)(22,37)(24,38)(25,36)(27,31)(29,34)(30,35)(32,40)(33,39);(9,11)(10,16)(12,22)(13,27)(14,30)(15,24)(17,31)(18,32)(19,35)(20,33)(21,26)(23,28)(25,39)(29,40)(34,38)(36,37);(2,5)(6,8)(9,12,15,14)(10,17,20,18)(11,22,24,30)(13,25,29,28)(16,31,33,32)(19,26,36,34)(21,37,38,35)(23,27,39,40);(9,12,15,14)(10,18,20,17)(11,22,24,30)(13,25,29,28)(16,32,33,31)(19,34,36,26)(21,35,38,37)(23,27,39,40)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=ch,d^a=di,e^a=eg,g^a=gi,e^b=eg,g^b=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72285609, #72285673, #72285724, … |
| maximal normal subgroup of | #72285609, #72285673, #72285724, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388464);