Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9938 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#392 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1927 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(6,8)(9,10)(11,18)(12,16)(13,21)(14,17)(15,19)(20,24)(22,23);(1,2)(3,8)(4,6)(5,7)(9,11)(10,15)(12,19)(13,22)(14,20)(16,24)(17,23)(18,21);(3,7)(5,8)(15,23)(18,24);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,24,23,18)Polycyclic presentation:
2^9|d^2=hi,e^2=h,b^a=be,c^a=cf,e^a=eh,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239697, #72249689, #72257694, … |
| maximal normal subgroup of | #72239697, #72249689, #72257694, … |
| maximal quotient of | #72264033, #72264045, #72264064, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 142126);