Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3341 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1826 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,3)(5,6,9,12)(7,10,13,16)(8,11,15,18)(14,17,19,20)(21,22,24,28)(23,30,31,26)(25,27,33,29)(32,35,36,34);(1,2,4,3)(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23)(22,26)(24,31)(25,32)(27,34)(28,30)(29,35)(33,36);(1,3,4,2)(7,13)(10,16)(14,19)(17,20)(22,27)(25,33)(26,34)(28,29)(30,35)(32,36);(5,8)(6,11)(7,14)(9,15)(10,17)(12,18)(13,19)(16,20)(23,31)(26,30)(32,36)(34,35);(8,15)(11,18)(14,19)(17,20)(23,31)(26,30)(32,36)(34,35)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,c^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ei,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6473449);