Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #272923 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1019 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)(7,15,18,11)(8,17)(9,16,20,13)(10,14)(12,19)(21,22)(23,30)(24,32)(25,34)(26,29)(27,31)(28,35)(33,36);(1,2)(3,4)(5,7,10,18)(6,11,14,15)(8,16,19,13)(9,17,20,12)(21,23,26,24)(22,27,29,36)(25,33,35,31)(28,32,34,30);(2,4)(5,8)(6,12)(7,16)(9,11)(10,19)(13,18)(14,17)(15,20);(5,9)(6,13)(7,17)(8,11)(10,20)(12,18)(14,16)(15,19)(21,24,26,23)(22,27,29,36)(25,31,35,33)(28,32,34,30)Polycyclic presentation:
2^9|a^2=fhi,b^2=h,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,e^c=ehi,e^d=ei,f^d=fhi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 180233);