Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480274 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#519 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1143 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,9,14,10,15,23,33)(7,11,18,27,16,25,34,36)(8,13,20,30,21,24,22,32)(12,28,29,17,26,35,31,19);(1,2,3,4)(5,7)(6,11)(8,17)(9,18)(10,16)(12,24)(13,26)(14,27)(15,25)(19,21)(20,35)(22,28)(23,34)(29,32)(30,31)(33,36);(5,7)(6,12)(8,19)(9,18)(10,16)(11,24)(13,25)(14,31)(15,26)(17,21)(20,28)(22,35)(23,34)(27,30)(29,33)(32,36);(7,16)(11,25)(12,26)(17,19)(18,34)(27,36)(28,35)(29,31)Polycyclic presentation:
2^9|a^2=g,b^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,d^c=di,f^c=fi,g^c=ghi,g^f=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400924);