Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #184831 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C22×#534 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×D4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
5154 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(9,10)(11,14)(12,19,20,15)(13,16)(17,23,24,21)(18,22);(2,4)(5,6)(7,8)(9,11,13,18)(10,14,16,22)(12,17,20,24)(15,21,19,23);(6,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,24)(22,23);(2,4)(11,18)(14,22)(17,24)(21,23)Polycyclic presentation:
2^9|a^2=h,b^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72225690, #72240675, #72249356, … |
| maximal normal subgroup of | #72225690, #72240675, #72249356, … |
| maximal quotient of | #72225649, #72225651, #72225675, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266630);