Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454375 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
962 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(6,9)(10,12)(13,14,17,19)(15,20,16,18)(21,22)(23,32)(24,35)(25,29)(26,30)(27,31)(28,33)(34,36);(5,6,8,9)(7,10,11,12)(21,23,25,33,27,24,26,34)(22,28,30,32,31,36,29,35);(5,7,8,11)(6,10,9,12)(13,15,17,16)(14,18,19,20)(22,29)(23,33)(24,34)(25,26)(30,31)(32,35);(2,4)(7,11)(10,12)(13,16,17,15)(14,18,19,20)(21,24,25,34,27,23,26,33)(22,28,30,32,31,36,29,35)Polycyclic presentation:
2^9|a^2=fg,b^2=e,c^2=efi,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 370760);