Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #267947 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1169 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)(3,7,4,6)(5,8)(9,10,13,17,14,12,23,16)(11,18,20,19,21,15,22,24);(1,3,5,4)(2,6,8,7)(9,11,14,21)(10,15,12,18)(13,20,23,22)(16,19,17,24);(1,2)(3,6)(4,7)(5,8)(9,12)(10,14)(11,19)(13,17)(15,22)(16,23)(18,20)(21,24);(1,4,5,3)(2,7,8,6)(9,13,14,23)(10,16,12,17)(11,20,21,22)(15,19,18,24)Polycyclic presentation:
2^9|a^2=f,b^2=hi,d^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^b=fh,g^b=gi,e^c=eh,f^c=fi,g^c=gi,f^d=fh50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 175229);