Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456670 |
| 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 | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
952 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)(7,12,17,27)(8,21,23,29)(9,13)(10,16)(11,15)(14,22,30,24)(18,32)(19,28,20,25)(26,34)(31,35)(33,36);(5,7,9,19,11,17,10,20)(6,12,16,25,15,27,13,28)(8,18,22,33,23,31,24,34)(14,26,29,35,30,36,21,32);(6,13)(7,17)(9,10)(12,25)(14,29)(15,16)(21,30)(22,24)(26,35)(27,28)(32,36)(33,34);(2,4)(5,8)(6,14)(7,18)(9,22)(10,24)(11,23)(12,26)(13,21)(15,30)(16,29)(17,31)(19,33)(20,34)(25,35)(27,36)(28,32)Polycyclic presentation:
2^9|a^2=g,b^2=f,f^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,f^c=fi,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, 367763);