Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #333055 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
781 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)(9,10,13,16,14,17,24,23)(11,15,19,22,21,12,20,18);(1,2,5,6)(3,7,8,4)(9,10)(11,18)(12,19)(13,23)(14,17)(15,20)(16,24)(21,22);(1,3,5,8)(2,7,6,4)(9,11,14,21)(10,15,17,12)(13,19,24,20)(16,22,23,18);(1,4)(2,3)(5,7)(6,8)(9,12)(10,11)(13,22)(14,15)(16,20)(17,21)(18,24)(19,23)Polycyclic presentation:
2^9|a^2=ei,b^2=i,c^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=defh,f^a=fi,c^b=cg,d^b=dei,e^b=eh,g^b=gh,f^c=fi,g^c=gh,e^d=eh,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 240299);