Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213818 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1389 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,16,19,11)(8,20,22,30)(9,12)(10,14)(13,21,29,36)(15,24)(17,31)(18,27,23,25)(26,32)(28,35)(33,34);(1,2)(3,4)(5,7)(6,11)(8,17)(9,23)(10,19)(12,27)(13,28)(14,16)(15,25)(18,24)(20,34)(21,35)(22,33)(26,29)(30,31)(32,36);(2,4)(6,12,14,15)(11,25,16,27)(13,20,29,30)(26,34,28,31);(5,8,10,22)(6,13)(7,17,19,33)(9,21,24,36)(11,26)(12,20)(14,29)(15,30)(16,28)(18,32,23,35)(25,34)(27,31)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290946);