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×#3560 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2362 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)(21,22,25,30,26,31,24,29)(23,28,32,27,34,35,33,36);(2,4)(5,7)(6,8)(9,13)(10,14)(11,15)(12,16)(17,19)(18,20)(21,23)(22,27)(24,32)(25,33)(26,34)(28,30)(29,35)(31,36);(5,8)(6,7)(9,15)(10,16)(11,13)(12,14)(17,20)(18,19)(21,23)(22,28)(24,33)(25,32)(26,34)(27,30)(29,36)(31,35);(5,9,10,18)(6,11,12,17)(7,13,14,20)(8,15,16,19)(21,24,26,25)(22,29,31,30)(23,32,34,33)(27,35,36,28)Polycyclic presentation:
2^9|a^2=g,d^2=hi,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, 93239);