Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476613 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2826 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2018 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1036 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,25,11,14,10,15)(7,16,19,12,20,27,18,28)(8,13,22,30,24,32,23,31)(17,29,21,35,33,36,34,26);(1,2)(3,4)(5,7,9,19,11,20,10,18)(6,12,15,16,14,28,25,27)(8,21,23,17,24,34,22,33)(13,29,30,35,32,36,31,26);(2,4)(5,8,9,23,11,24,10,22)(6,13,15,30,14,32,25,31)(7,17,19,34,20,33,18,21)(12,26,16,29,28,35,27,36);(6,14)(7,18)(8,22)(9,10)(12,27)(13,30)(16,28)(17,33)(19,20)(23,24)(26,35)(31,32)Polycyclic presentation:
2^9|a^2=ef,b^2=e,c^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=def,g^b=gi,h^b=hi,d^c=de,f^c=fi,g^c=gi,h^c=hi,e^d=ei,g^d=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383809);