Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31418 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1259 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)(7,17)(8,21,23,13)(9,24,22,36)(10,26)(11,16)(12,19)(14,29,31,25)(15,30)(18,35,28,32)(20,34)(27,33);(1,2)(3,4)(5,7)(6,12)(8,22)(9,23)(10,27)(11,20)(13,14)(15,34)(16,33)(17,26)(18,25)(19,30)(21,31)(24,32)(28,29)(35,36);(2,4)(5,8,10,28,11,23,30,18)(6,13,15,32,16,21,26,35)(7,9,19,29,20,22,27,25)(12,31,17,24,33,14,34,36);(5,9)(6,14)(7,18)(8,19)(10,29)(11,22)(12,32)(13,34)(15,36)(16,31)(17,21)(20,28)(23,27)(24,26)(25,30)(33,35)Polycyclic presentation:
2^9|a^2=h,c^2=eh,e^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295590);