Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7501009 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3135 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6,12,30,11,16,10,15)(7,17,29,13,20,34,21,32)(8,9,24,27,14,28,23,26)(18,35,22,31,36,25,33,19);(5,7)(6,13)(8,22)(9,25)(10,29)(11,20)(12,21)(14,33)(15,34)(16,32)(17,30)(18,24)(19,26)(23,36)(27,31)(28,35);(2,4)(5,8,11,14)(6,9,16,28)(7,18,20,36)(10,23,12,24)(13,31,32,19)(15,26,30,27)(17,35,34,25)(21,33,29,22);(2,4)(5,9)(6,14)(7,19)(8,16)(10,27)(11,28)(12,26)(13,18)(15,23)(17,22)(20,31)(21,35)(24,30)(25,29)(32,36)(33,34);(7,20)(13,32)(17,34)(18,36)(19,31)(21,29)(22,33)(25,35)Polycyclic presentation:
2^9|a^2=fi,c^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,f^b=fi,h^b=hi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408210);