Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31077 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×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×#3018 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1934 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,17,12,18,10,16)(7,19,21,13,22,32,20,33)(8,23,25,14,27,35,26,31)(9,15,24,36,28,30,29,34);(1,2)(3,4)(5,7)(6,13)(8,24)(9,25)(10,21)(11,20)(12,22)(14,34)(15,23)(16,32)(17,19)(18,33)(26,28)(27,29)(30,35)(31,36);(2,4)(5,8)(6,14)(7,9)(10,25)(11,26)(12,27)(13,30)(15,33)(16,35)(17,23)(18,31)(19,34)(20,24)(21,29)(22,28)(32,36);(5,9)(6,15)(7,8)(10,29)(11,24)(12,28)(13,31)(14,33)(16,34)(17,36)(18,30)(19,35)(20,26)(21,25)(22,27)(23,32)Polycyclic presentation:
2^9|a^2=h,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=defi,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=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290261);