Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417661 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1914 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,12,28,11,15,10,14)(7,16,20,8,19,29,18,22)(9,23,25,31,26,32,27,13)(17,21,24,30,34,36,35,33);(1,2)(3,4)(5,7,11,19)(6,8,15,22)(9,24,26,35)(10,20,12,18)(13,30,31,33)(14,29,28,16)(17,27,34,25)(21,32,36,23);(2,4)(5,8)(6,7)(9,21)(10,16)(11,22)(12,29)(13,17)(14,18)(15,19)(20,28)(23,24)(25,30)(26,36)(27,33)(31,34)(32,35);(5,9)(6,13)(7,17)(8,21)(10,27)(11,26)(12,25)(14,32)(15,31)(16,33)(18,35)(19,34)(20,24)(22,36)(23,28)(29,30)Polycyclic presentation:
2^9|a^2=ef,b^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^b=hi,h^c=hi,e^d=ei,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326022);