Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232128 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2380 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1037 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,11,17,28)(8,19,22,30)(9,15)(10,14)(12,23,31,36)(13,21)(16,32)(18,26,24,27)(20,34)(25,35)(29,33);(5,7)(6,11)(8,20)(9,17)(10,24)(12,29)(13,26)(14,27)(15,28)(16,23)(18,21)(19,34)(22,35)(25,30)(31,32)(33,36);(1,2,4,3)(5,8,21,36,9,22,10,23)(6,12,14,19,15,31,13,30)(7,16,24,35,17,33,18,20)(11,25,26,32,28,34,27,29);(1,3,4,2)(6,13,15,14)(7,17)(11,26,28,27)(12,30,31,19)(16,33)(18,24)(20,35)(25,32,34,29)Polycyclic presentation:
2^9|a^2=h,c^2=ef,d^2=fgi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 294831);