Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #428059 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
889 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)(6,11)(7,15)(8,19)(9,20)(12,25)(13,23)(16,28)(17,31)(18,26)(27,36)(32,35)(33,34);(5,6,9,14,10,11,20,24)(7,16,15,25,18,12,26,28)(8,13,21,29,19,23,22,30)(17,32,34,36,33,27,31,35);(5,7,9,15,10,18,20,26)(6,12,24,25,11,16,14,28)(8,17,22,31,19,33,21,34)(13,27,29,35,23,32,30,36);(2,4)(5,8,20,21,10,19,9,22)(6,13,14,30,11,23,24,29)(7,17,26,34,18,33,15,31)(12,27,28,36,16,32,25,35)Polycyclic presentation:
2^9|a^2=egh,b^2=f,c^2=ef,d^2=efi,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,e^b=ei,h^b=hi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 359725);