Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24378 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#449 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1071 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,17)(8,20)(9,14)(10,15)(11,16)(12,18)(13,19)(21,22,25,29,26,30,34,36)(23,27,31,24,28,33,35,32);(1,2,3,4)(5,7,9,18,11,19,10,8)(6,12,15,17,16,20,14,13)(21,23,25,31,26,28,34,35)(22,27,29,24,30,33,36,32);(5,8,10,19,11,18,9,7)(6,13,14,20,16,17,15,12)(21,24)(22,28)(23,30)(25,33)(26,32)(27,34)(29,35)(31,36);(5,9,11,10)(6,14,16,15)(7,18,19,8)(12,13,20,17)(23,28)(24,32)(27,33)(31,35)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=f,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=di,d^c=di116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 111598);