Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806903 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1749 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6,9,12)(7,13,15,10)(8,16,17,11)(14,18,20,19);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,22,27,31)(23,30,33,36)(24,29,26,35)(25,34,32,28);(2,4)(21,23,24,25,27,33,26,32)(22,28,35,36,31,34,29,30);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(21,24,27,26)(22,29,31,35)(23,32,33,25)(28,36,34,30);(7,15)(10,13)(14,20)(18,19)(21,25,24,33,27,32,26,23)(22,30,35,28,31,36,29,34)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=hi,d^2=fi,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6716090);