Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480268 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#520 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1165 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)(8,12)(9,22,23,13)(10,14)(11,17,25,32)(16,29)(18,30)(19,26)(20,34)(21,36,33,28)(24,31,27,35);(1,2,3,4)(8,19)(12,26)(16,20)(21,33)(24,27)(28,36)(29,34)(31,35);(5,7,10,18)(6,11,14,25)(8,20,19,16)(9,17,23,32)(12,27,26,24)(13,15,22,30)(21,35,33,31)(28,34,36,29);(5,8,10,19)(6,12,14,26)(7,16,18,20)(9,21,23,33)(11,24,25,27)(13,28,22,36)(15,29,30,34)(17,31,32,35)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=i,d^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,d^c=di,g^c=gh,g^f=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400900);