Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #248503 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3496 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1237 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(3,7)(4,8)(9,10)(11,19)(12,16)(13,17)(14,18)(15,20)(21,24)(22,23);(1,2,5,3)(4,7,8,6)(9,11,14,21)(10,15,18,23)(12,20,13,22)(16,24,17,19);(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,19,23,24);(1,3,5,2)(4,6,8,7)(10,17)(12,13)(15,19)(16,18)(20,22)(23,24)Polycyclic presentation:
2^9|b^2=h,c^2=i,d^2=hi,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,e^b=eh,f^b=fi,d^c=di,e^d=eh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72263900, #72263906, #72263908, … |
| maximal normal subgroup of | #72263900, #72263906, #72263908, … |
| maximal quotient of | #72264078, #72264219 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 155742);