Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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×#3345 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1830 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6,9,12)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22,25,30,26,29,27,23)(24,32,35,31,34,28,36,33);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23)(22,27)(24,33)(25,29)(26,30)(28,35)(31,34)(32,36);(2,4)(5,8,9,16)(6,11,12,20)(7,13,14,15)(10,17,18,19)(21,24)(22,28)(23,31)(25,35)(26,34)(27,36)(29,32)(30,33);(8,16)(11,20)(13,15)(17,19)(21,22,27,23,26,29,25,30)(24,32,36,33,34,28,35,31);(6,12)(7,14)(11,20)(13,15)(22,29)(23,30)(24,34)(35,36)Polycyclic presentation:
2^9|a^2=fhi,c^2=hi,d^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72257063, #72257127, #72257193, … |
| maximal normal subgroup of | #72257063, #72257127, #72257193, … |
| maximal quotient of | #72265460 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7011989);