Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453981 |
| 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 | |
1272 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)(5,6,8,9)(7,10,11,12)(21,22,23,28,27,32,26,31)(24,35,34,29,25,36,33,30);(6,9)(10,12)(13,14)(15,18)(16,17)(19,20)(21,23,27,26)(22,28,32,31)(24,33,25,34)(29,35,30,36);(2,4)(5,7)(6,10)(8,11)(9,12)(14,17)(18,20)(21,24,27,25)(22,29,32,30)(23,33,26,34)(28,35,31,36);(7,11)(10,12)(13,15,16,19)(14,18,17,20)(21,25,27,24)(22,30,32,29)(23,34,26,33)(28,36,31,35)Polycyclic presentation:
2^9|a^2=efi,b^2=i,c^2=i,d^2=gi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,f^c=fi,h^c=hi,f^d=fi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 376720);