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 | C23×D4 | 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 | |
1186 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)(5,6,9,14,10,15,22,29)(7,11,18,26,16,24,31,35)(8,13,20,28,19,12,21,27)(17,25,32,36,30,23,33,34);(1,2,3,4)(5,7)(6,11)(8,17)(9,18)(10,16)(12,23)(13,25)(14,26)(15,24)(19,30)(20,32)(21,33)(22,31)(27,34)(28,36)(29,35);(6,12)(8,19)(11,23)(13,15)(14,28)(17,30)(20,21)(24,25)(26,36)(27,29)(32,33)(34,35);(7,16)(11,24)(17,30)(18,31)(23,25)(26,35)(32,33)(34,36)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,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, 400920);