Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419041 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2424 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2228 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,10,16,12,9,11,15)(7,17,19,13,21,30,20,28)(8,22,26,24,27,32,25,14)(18,34,35,31,36,29,23,33);(1,2,3,4)(5,7,10,19,12,21,11,20)(6,13,15,17,9,28,16,30)(8,23,25,36,27,35,26,18)(14,31,22,29,24,33,32,34);(5,8)(6,14)(7,18)(9,24)(10,25)(11,26)(12,27)(13,29)(15,22)(16,32)(17,33)(19,23)(20,35)(21,36)(28,34)(30,31);(2,4)(5,9,12,6)(7,17,21,30)(8,24,27,14)(10,15,11,16)(13,20,28,19)(18,33,36,31)(22,26,32,25)(23,29,35,34)Polycyclic presentation:
2^9|a^2=f,b^2=e,d^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi,e^d=ei,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326238);