Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30437 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| 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 | |
1983 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6)(7,17)(8,22)(9,15)(10,14)(11,16)(12,20)(13,23)(18,29)(19,28)(21,27)(24,31)(25,30)(26,35)(32,34)(33,36);(5,7,11,21)(6,12,16,28)(8,18,24,35)(9,19,10,20)(13,29,30,26)(14,27,15,17)(22,32,31,36)(23,33,25,34);(5,8)(6,13)(7,18)(9,23)(10,25)(11,24)(12,26)(14,22)(15,31)(16,30)(17,32)(19,33)(20,34)(21,35)(27,36)(28,29);(7,19)(9,10)(12,27)(14,15)(17,28)(18,33)(20,21)(22,31)(23,25)(26,36)(29,32)(34,35)Polycyclic presentation:
2^9|a^2=h,b^2=i,e^2=i,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,e^b=ei,f^b=fi,f^c=fi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290362);