Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417581 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1370 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(6,9)(10,12)(13,14,16,18)(15,17,19,20)(21,22,25,30)(23,28,26,27)(24,32,34,35)(29,33,36,31);(5,6)(7,10)(8,9)(11,12)(13,15)(14,17)(16,19)(18,20)(21,23,25,26)(22,27,30,28)(24,33,34,31)(29,35,36,32);(2,4)(5,7,8,11)(6,10,9,12)(15,19)(17,20)(22,28)(23,26)(27,30)(29,35)(31,33)(32,36);(6,9)(10,12)(21,24,25,34)(22,29,30,36)(23,31,26,33)(27,32,28,35)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=ei,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 339852);