Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31077 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×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 | #2782 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
803 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,16,12,17,11,15)(7,18,20,13,22,28,21,9)(8,23,26,14,27,32,29,33)(19,25,24,34,35,30,36,31);(1,2)(3,4)(5,7,11,21,12,22,10,20)(6,13,16,28,17,9,15,18)(8,24,26,35,27,36,29,19)(14,31,23,30,33,34,32,25);(2,4)(5,8,10,26,12,27,11,29)(6,14,16,32,17,33,15,23)(7,19,20,24,22,35,21,36)(9,25,18,34,13,30,28,31);(5,9)(6,7)(8,25)(10,18)(11,28)(12,13)(14,19)(15,21)(16,20)(17,22)(23,36)(24,32)(26,34)(27,30)(29,31)(33,35)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=h,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,f^c=fi,g^c=gi,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, 295895);