Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484095 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| prime radical | C2×#508 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1451 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
439 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,7,5,3,6,8)(9,10,15,20)(11,22)(12,27,28,19)(13,18,29,21)(14,17,31,36)(16,32)(23,35)(24,39)(25,37)(26,33)(30,38)(34,40);(9,11,15,26)(10,16,20,35)(12,23,28,32)(13,24,29,25)(14,30,31,40)(17,33,36,22)(18,34,21,38)(19,37,27,39);(1,3,4,8,5,2,6,7)(9,12,13,14,15,28,29,31)(10,17,21,27,20,36,18,19)(11,23,24,30,26,32,25,40)(16,33,38,39,35,22,34,37);(2,3)(7,8)(9,13,15,29)(10,18,20,21)(11,24,26,25)(12,14,28,31)(16,34,35,38)(17,19,36,27)(22,39,33,37)(23,30,32,40)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fgi,d^2=i,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cg,d^a=dh,e^a=eg,g^a=gi,e^b=egi,f^b=fi,g^b=gi,d^c=dh,e^c=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72286070, #72286241, #72286498, … |
| maximal normal subgroup of | #72286070, #72286241, #72286498, … |
| maximal quotient of | #72343386, #72343424, #72343529, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391298);