Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24384 |
| 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 | C22×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1060 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)(7,16)(8,12)(9,14)(10,19)(11,15)(13,17)(18,20)(21,22,25,29,26,30,24,27)(23,31,34,36,32,35,33,28);(1,2,3,4)(5,7,10,8,11,17,9,18)(6,12,14,16,15,20,19,13)(22,27,30,29)(23,32)(28,31,36,35)(33,34);(5,8,10,17,11,18,9,7)(6,13,14,12,15,16,19,20)(21,23,24,33,26,32,25,34)(22,28,29,31,30,36,27,35);(5,9,11,10)(6,14,15,19)(7,8,17,18)(12,13,20,16)(21,24,26,25)(22,27,30,29)(23,33,32,34)(28,35,36,31)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,c^b=ch,d^b=dhi,d^c=dhi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 111620);