Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18446 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3561 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,8,14,16)(9,17)(10,12)(11,18)(13,19,20,15)(21,22,25,29,26,28,24,34)(23,30,32,27,33,36,31,35);(2,4)(5,7)(6,8)(9,13)(10,14)(11,15)(12,16)(17,19)(18,20)(21,23,26,33)(22,27,28,35)(24,31,25,32)(29,36,34,30);(5,8)(6,7)(9,15)(10,16)(11,13)(12,14)(17,20)(18,19)(22,28)(24,25)(27,35)(31,32);(5,9,10,18)(6,11,12,17)(7,13,14,20)(8,15,16,19)(21,24,26,25)(22,29,28,34)(23,31,33,32)(27,36,35,30)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fh,e^b=ei,f^b=fh,g^b=ghi,d^c=di,e^c=eh,g^c=gh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 93235);