Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C42 | C24 |
| commutator, socle | C23 | C22×C42 |
| prime radical | C22×D4 | C42 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
898 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,15,12,11,14,16)(17,18,21,19,20,23,22,24);(1,3)(2,5)(4,8)(6,7)(10,11)(15,16)(17,18,22,24,20,23,21,19);(3,8)(5,7)(9,11,13,16,12,10,14,15)(17,19,21,23,20,24,22,18);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,f^2=h,g^2=i,b^a=be,c^a=ci,d^a=deh,c^b=cei152 conjugacy classes — view the full character table.
| maximal subgroup of | #72343135, #72343147, #72343201 |
| maximal normal subgroup of | #72343135, #72343147, #72343201 |
| maximal quotient of | #72343638, #72343659 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 91846);