Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420455 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15278 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1668 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,4,7,11,6,18)(3,12,14,9,15,16,23,26)(5,19,24,30,22,10,21,17)(13,27,20,25,28,29,32,31);(1,3)(2,9)(4,16)(5,20)(6,23)(7,15)(8,14)(10,27)(11,26)(12,18)(13,21)(17,25)(19,29)(22,32)(24,28)(30,31);(1,4,6,11,7,18,8,2)(3,12,14,26,15,16,23,9)(5,17,21,19,22,30,24,10)(13,27,20,31,28,29,32,25);(1,5,6,21,7,22,8,24)(2,10,4,17,11,19,18,30)(3,13,14,20,15,28,23,32)(9,25,12,27,26,31,16,29)Polycyclic presentation:
2^9|a^2=ef,c^2=e,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,f^c=fi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327651);