Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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×#3301 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1827 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,14)(8,12)(9,18)(10,13)(11,17)(15,20)(16,19)(21,22,25,27)(23,31,32,28)(24,29,34,36)(26,30,35,33);(5,7)(6,11)(8,15)(9,16)(10,17)(12,19)(13,14)(18,20)(22,27)(23,29)(24,31)(26,35)(28,34)(32,36);(1,2,3,4)(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,19,14,20)(21,23)(22,28)(24,33)(25,32)(26,36)(27,31)(29,35)(30,34);(5,9,10,8)(6,12,13,18)(7,16,17,15)(11,19,14,20)(22,27)(23,32)(29,36)(30,33);(7,17)(11,14)(15,16)(19,20)(21,24,25,34)(22,29,27,36)(23,33,32,30)(26,28,35,31)Polycyclic presentation:
2^9|a^2=i,c^2=fg,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6591555);