Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232138 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2380 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3153 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1618 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,11,16,12,17,10,18)(7,19,22,13,23,31,28,30)(8,15,9,26,25,14,27,33)(20,32,21,29,35,34,24,36);(1,2)(3,4)(5,7)(6,13)(8,24)(9,20)(10,28)(11,22)(12,23)(14,32)(15,34)(16,31)(17,30)(18,19)(21,25)(26,36)(27,35)(29,33);(5,8,11,27,12,25,10,9)(6,14,16,26,17,15,18,33)(7,20,28,21,23,35,22,24)(13,29,19,34,30,36,31,32);(1,3,4,2)(5,9,10,25,12,27,11,8)(6,15,16,33,17,14,18,26)(7,21,22,20,23,24,28,35)(13,29,19,34,30,36,31,32)Polycyclic presentation:
2^9|a^2=h,c^2=gi,d^2=e,e^2=i,g^2=i,h^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,g^b=gi,h^c=hi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293233);