Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #464719 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2631 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
799 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,9,12)(7,17)(8,22,23,30)(10,15,11,16)(13,20)(14,25,27,24)(18,26)(19,33)(21,29)(28,35)(31,34)(32,36);(6,12)(7,18,21,20)(13,17,26,29)(14,27)(15,16)(19,32,35,34)(22,30)(28,31,33,36);(5,7,11,18,9,21,10,20)(6,13,16,17,12,26,15,29)(8,19,25,34,23,35,24,32)(14,28,22,36,27,33,30,31);(2,4)(5,8,11,24,9,23,10,25)(6,14,15,22,12,27,16,30)(7,19,18,32,21,35,20,34)(13,28,29,36,26,33,17,31)Polycyclic presentation:
2^9|a^2=e,b^2=ei,c^2=ef,d^2=eg,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 377060);