Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480263 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1195 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,15,18,30)(8,12)(9,22,24,13)(10,14)(11,17)(16,31,19,33)(20,35,36,28)(21,29)(23,26)(25,32)(27,34);(1,2,3,4);(5,7,10,18)(6,11,14,26)(8,19,21,16)(9,23,24,17)(12,27,29,25)(13,30,22,15)(20,32,36,34)(28,31,35,33);(5,8,10,21)(6,12,14,29)(7,16,18,19)(9,20,24,36)(11,25,26,27)(13,28,22,35)(15,31,30,33)(17,32,23,34)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=i,d^2=i,f^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,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, 400944);