Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31076 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,𝔽4) |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2684 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1064 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6,10,15,11,16,24,33)(7,17,20,32,22,31,21,12)(8,23,19,35,26,34,25,13)(9,27,28,30,29,14,18,36);(5,7)(6,12)(8,9)(10,21)(11,22)(13,14)(15,31)(16,32)(17,33)(18,19)(20,24)(23,30)(25,28)(26,29)(27,35)(34,36);(5,8,24,19,11,26,10,25)(6,13,33,23,16,35,15,34)(7,18,20,29,22,28,21,9)(12,30,17,27,32,36,31,14);(5,9,24,18,11,29,10,28)(6,14,33,30,16,27,15,36)(7,19,20,26,22,25,21,8)(12,23,17,35,32,34,31,13)Polycyclic presentation:
2^9|a^2=h,c^2=efg,d^2=efg,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295272);