Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456658 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
918 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)(5,6,11,16,10,15,26,32)(7,12,20,28,19,30,21,29)(8,13,23,14,9,24,25,33)(17,27,22,31,34,36,18,35);(5,7,10,19)(6,12)(8,22,9,18)(11,21,26,20)(13,31)(14,27)(15,30)(16,29)(17,25,34,23)(24,35)(28,32)(33,36);(5,8,10,9)(6,13,15,24)(7,17,19,34)(11,25,26,23)(12,27,30,36)(14,16,33,32)(18,21,22,20)(28,35,29,31);(2,4)(5,9,10,8)(6,14,15,33)(7,18,19,22)(11,23,26,25)(12,27,30,36)(13,32,24,16)(17,20,34,21)(28,35,29,31)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=i,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,d^c=de,g^c=gi,h^c=hi,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, 367760);