Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454986 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1104 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)(5,6,11,17)(7,18,21,29)(8,23,26,13)(9,16,10,15)(12,22,14,20)(19,34,36,30)(24,28,35,32)(25,31,27,33);(5,7)(6,12)(8,19,26,36)(9,22)(10,20)(11,21)(13,28,23,32)(14,17)(15,29)(16,18)(24,27,35,25)(30,31,34,33);(5,8,9,25,11,26,10,27)(6,13,15,31,17,23,16,33)(7,19,20,35,21,36,22,24)(12,28,18,30,14,32,29,34);(2,4)(5,7)(6,14)(8,24)(9,22)(10,20)(11,21)(12,17)(13,30)(15,18)(16,29)(19,27)(23,34)(25,36)(26,35)(28,31)(32,33)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,d^c=de,f^c=fi,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363035);