Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32941 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C4×#121 |
| prime radical | C2×#3067 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1800 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,12,18,28)(8,20)(9,24)(10,16)(11,15)(13,21)(14,19,27,25)(17,29,31,36)(22,30)(23,34)(26,35,33,32);(1,2,3,4)(5,7)(6,12,15,28)(8,17)(9,19)(10,25)(11,18)(13,26,30,33)(14,24,27,16)(20,29,34,36)(21,32)(22,35)(23,31);(5,8,9,21,11,23,10,22)(6,13,16,20,15,30,24,34)(7,17,19,32,18,31,25,35)(12,26,14,29,28,33,27,36);(5,7)(6,14)(8,17)(9,19)(10,25)(11,18)(12,24)(13,29)(15,27)(16,28)(20,33)(21,32)(22,35)(23,31)(26,34)(30,36)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,e^b=ei,g^b=gi,h^b=hi,e^d=ei,g^d=gi,h^d=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300290);