Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #185503 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3484 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
4174 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10,13,18,14,19,12,17)(11,16,20,15,22,23,21,24);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,20)(13,21)(14,22)(16,18)(17,23)(19,24);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,21)(13,20)(14,22)(15,18)(17,24)(19,23);(9,12,14,13)(10,17,19,18)(11,20,22,21)(15,23,24,16)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=ei,g^b=ghi,d^c=di,g^c=gh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72235084, #72238182, #72238184, … |
| maximal normal subgroup of | #72235084, #72238182, #72238184, … |
| maximal quotient of | #72314099, #72314578, #72316705, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92879);