Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9914 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #120 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C2×#2656 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
2018 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(9,10)(11,12,15,16)(13,14)(17,18)(19,22)(20,23)(21,24);(2,4)(5,6)(7,8)(9,11)(10,12)(13,15)(14,16)(18,22)(19,20)(23,24);(6,8)(9,12)(10,11)(13,16)(14,15)(18,23)(19,20)(22,24)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=dh,e^c=ei,g^c=ghi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72217673, #72238061, #72267754, … |
| maximal normal subgroup of | #72217673, #72238061, #72267754, … |
| maximal quotient of | #72216328, #72238063, #72238064, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18462);