Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456658 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×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 | |
1374 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,16)(7,13)(8,23,9,14)(11,18,12,17)(15,26,32,25)(19,28)(20,31)(21,30)(22,33)(24,35)(27,29)(34,36);(1,2)(3,4)(5,7,10,21)(6,13)(8,24,9,20)(11,27,12,22)(14,31)(15,28)(16,30)(17,33)(18,29)(19,26,34,25)(23,35)(32,36);(2,4)(5,8,10,9)(6,14,16,23)(7,19,21,34)(11,26,12,25)(13,28,30,36)(15,17,32,18)(20,27,24,22)(29,35,33,31);(5,9,10,8)(6,15,16,32)(7,20,21,24)(11,25,12,26)(13,28,30,36)(14,18,23,17)(19,22,34,27)(29,35,33,31)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=i,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,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, 363857);