Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395944 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#396 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C22×Q8 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
786 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,6)(3,7,4,8)(9,10,15,19,14,18,13,20)(11,21,24,32,25,33,23,16)(12,26,28,35,30,36,29,17)(22,37,27,34,39,31,38,40);(2,6)(7,8)(9,11,15,24,14,25,13,23)(10,16,20,33,18,32,19,21)(12,27,29,22,30,38,28,39)(17,34,26,31,35,40,36,37);(1,3,5,4)(2,7,6,8)(9,12,13,28,14,30,15,29)(10,17,19,36,18,35,20,26)(11,22,23,27,25,39,24,38)(16,31,21,34,32,37,33,40);(1,4,5,3)(2,7,6,8)(10,18)(16,32)(17,35)(19,20)(21,33)(26,36)(31,37)(34,40)Polycyclic presentation:
2^9|a^2=efg,b^2=eh,c^2=e,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324173);