Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31414 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
823 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6,12,16,11,7,10,15)(8,13,21,29,23,17,22,19)(9,24,27,18,26,30,25,14)(20,32,33,31,35,36,34,28);(5,7,11,6)(8,19)(9,24,26,30)(10,15,12,16)(13,22)(14,25,18,27)(17,21)(20,31)(23,29)(28,35)(32,33)(34,36);(1,3,4,2)(5,8)(6,13,7,17)(9,20)(10,22)(11,23)(12,21)(14,28,18,31)(15,19,16,29)(24,32,30,36)(25,34)(26,35)(27,33);(5,9,11,26)(6,14,7,18)(8,20,23,35)(10,27,12,25)(13,28,17,31)(15,24,16,30)(19,32,29,36)(21,34,22,33)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=gh,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,g^a=gi,c^b=cg,d^b=de,e^b=ei,h^b=hi,e^c=ei,f^c=fi,e^d=ei,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 297784);