Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #300205 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2433 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #18620 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1079 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)(3,8)(4,7)(9,10)(11,19)(12,22)(13,16)(14,18)(15,20)(17,23)(21,24);(1,2,7,8,5,3,4,6)(9,11)(10,15)(12,14)(13,20)(16,19)(17,22)(18,24)(21,23);(2,3)(6,8)(9,12,14,20,13,21,23,11)(10,15,17,22,16,19,18,24);(1,3,4,8,5,2,7,6)(10,16)(15,19)(17,18)(22,24)Polycyclic presentation:
2^9|b^2=ehi,c^2=f,d^2=ei,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=dh,g^b=gi,d^c=dh,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 207451);