Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427438 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15369 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23402 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1275 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,8,19,24)(3,11,13,22,14,27,18,7)(4,9,16,26,17,23,20,28)(12,25,15,32,30,29,31,21);(1,3)(2,7)(4,15)(5,18)(6,14)(8,22)(9,25)(10,27)(11,24)(12,16)(13,19)(17,31)(20,30)(21,26)(23,29)(28,32);(2,8)(5,19)(9,23)(11,27)(13,18)(15,31)(16,20)(25,29);(1,4)(2,9,8,23)(3,12)(5,20)(6,17)(7,21,22,32)(10,28,24,26)(11,29,27,25)(13,31)(14,30)(15,18)(16,19)Polycyclic presentation:
2^9|a^2=e,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^b=hi,e^c=ei,e^d=ei,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 334634);