Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476609 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
701 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,12,20,15)(13,18)(14,17)(16,22,23,24)(19,21);(1,2)(3,6)(4,8)(5,7)(9,11,19,16,13,20,14,23)(10,15,17,22,18,12,21,24);(1,3,5,4)(2,6,7,8)(11,20)(12,15)(14,19)(17,21);(1,4,5,3)(2,6,7,8)(9,12,13,15)(10,16,18,23)(11,21,20,17)(14,24,19,22)Polycyclic presentation:
2^9|a^2=g,b^2=eh,c^2=f,d^2=fi,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=def,e^b=ei,g^b=gi,h^b=hi,d^c=de,h^c=hi,g^d=gi,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387023);