Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32696 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
711 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6,10,15,11,14,9,22)(7,12,16,27,18,26,19,24)(8,13,21,30,20,31,23,29)(17,25,34,35,33,28,32,36);(5,7)(6,12,14,26)(8,17,20,33)(9,16)(10,19)(11,18)(13,28)(15,24,22,27)(21,32,23,34)(25,31)(29,36)(30,35);(1,3,4,2)(7,16,18,19)(12,24,26,27)(17,32,33,34)(25,35,28,36);(5,8,9,21,11,20,10,23)(6,13,15,29,14,31,22,30)(7,17,19,34,18,33,16,32)(12,25,27,36,26,28,24,35)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=egh,d^2=g,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^d=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298372);