Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419895 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
978 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)(7,19,9,28)(8,23)(10,16)(11,18)(12,17)(13,21,15,22)(14,26)(20,29,24,30)(25,31)(27,32)(33,34,36,35);(5,7,11,22,12,9,10,21)(6,13,16,19,17,15,18,28)(8,24,26,34,27,20,25,35)(14,29,23,36,31,30,32,33);(5,8,12,27)(6,14,17,31)(7,20,9,24)(10,26,11,25)(13,29,15,30)(16,32,18,23)(19,33,28,36)(21,35,22,34);(2,4)(5,9,12,7)(6,15)(8,24,27,20)(10,21,11,22)(13,17)(14,30)(16,19)(18,28)(23,36)(25,34,26,35)(29,31)(32,33)Polycyclic presentation:
2^9|a^2=efg,b^2=f,c^2=i,d^2=efi,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 344166);