Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417581 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1188 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(6,9)(10,12)(13,14,16,18)(15,19,20,17)(21,22,27,33)(23,29,35,36)(24,30,34,28)(25,32,26,31);(5,6)(7,10)(8,9)(11,12)(13,15,16,20)(14,17,18,19)(21,23,25,34,27,35,26,24)(22,28,31,29,33,30,32,36);(5,7,8,11)(6,10,9,12)(14,18)(17,19)(21,24)(22,29)(23,26)(25,35)(27,34)(28,31)(30,32)(33,36);(2,4)(6,9)(10,12)(15,20)(17,19)(21,23,27,35)(22,30,33,28)(24,26,34,25)(29,32,36,31)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=efi,d^2=i,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^b=dei,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 342696);