Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419895 |
| 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 | |
1159 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,17,12,15,11,16)(7,13,22,31,20,33,21,32)(8,23,14,29,25,28,26,9)(18,27,24,30,35,19,34,36);(1,2,3,4)(5,7,12,20)(6,13)(8,24,25,34)(9,27)(10,21,11,22)(14,18,26,35)(15,33)(16,31)(17,32)(19,29)(23,36)(28,30);(2,4)(5,8,11,26,12,25,10,14)(6,9,16,28,15,29,17,23)(7,18,21,34,20,35,22,24)(13,30,32,27,33,36,31,19);(5,9)(6,14)(7,19)(8,16)(10,23)(11,28)(12,29)(13,18)(15,26)(17,25)(20,27)(21,30)(22,36)(24,31)(32,34)(33,35)Polycyclic presentation:
2^9|a^2=ef,b^2=efg,c^2=efi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 346547);