Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456501 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1448 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,15,11,16,9,14)(7,17,20,29,21,28,19,12)(8,22,24,31,26,30,25,13)(18,27,33,35,23,36,34,32);(1,2)(3,4)(5,7)(6,12,16,29)(8,23)(9,19)(10,20)(11,21)(13,27,31,36)(14,28,15,17)(18,26)(22,35,30,32)(24,34)(25,33);(2,4)(5,8)(6,13)(7,18)(9,24)(10,25)(11,26)(12,27)(14,22)(15,30)(16,31)(17,32)(19,33)(20,34)(21,23)(28,35)(29,36);(6,14)(8,24)(9,10)(12,28)(15,16)(17,29)(18,33)(19,20)(22,30)(23,34)(25,26)(32,35)Polycyclic presentation:
2^9|a^2=f,b^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,f^c=fi,e^d=ei,f^d=fi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363840);