Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29902 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2622 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1849 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,15,11,12,9,23)(7,13,16,27,18,25,24,29)(8,14,22,30,20,26,21,34)(17,28,31,35,19,32,33,36);(1,2)(3,4)(6,12)(7,16)(9,10)(13,27)(14,26)(17,31)(18,24)(19,33)(21,22)(25,29)(28,35)(32,36);(2,4)(5,7)(6,13)(8,19,20,17)(9,24)(10,16)(11,18)(12,25)(14,28,26,32)(15,27)(21,31,22,33)(23,29)(30,35,34,36);(5,8)(6,14)(7,17)(9,21)(10,22)(11,20)(12,26)(13,28)(15,30)(16,31)(18,19)(23,34)(24,33)(25,32)(27,35)(29,36)Polycyclic presentation:
2^9|a^2=h,c^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287467);