Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #245807 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1210 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(3,7)(4,8)(9,10)(11,19)(12,17)(13,16)(14,18)(15,21)(20,23)(22,24);(1,2)(3,5)(4,7)(6,8)(9,11)(10,15)(12,22)(13,21)(14,20)(16,23)(17,19)(18,24);(10,16)(12,13)(15,19)(17,18)(21,22)(23,24);(1,3,5,2)(4,6,8,7)(10,17)(11,20)(12,13)(15,23)(16,18)(19,24)Polycyclic presentation:
2^9|d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,c^b=cg,d^b=dh,e^b=eh,e^c=ei,f^c=fi,e^d=eh,f^d=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72257889, #72257895, #72257897, … |
| maximal normal subgroup of | #72257889, #72257895, #72257897, … |
| maximal quotient of | #72257930, #72264042 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 153057);