Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476613 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2826 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
663 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,11,16,12,15,10,9)(7,17,19,29,20,13,27,26)(8,21,24,14,23,31,25,32)(18,34,36,28,35,30,22,33);(5,7,10,19,12,20,11,27)(6,13,9,26,15,17,16,29)(8,22,25,18,23,36,24,35)(14,30,21,33,32,34,31,28);(5,8,10,25,12,23,11,24)(6,14,9,21,15,32,16,31)(7,18,19,36,20,35,27,22)(13,28,26,30,17,33,29,34);(2,4)(5,9,11,15,12,16,10,6)(7,13,27,26,20,17,19,29)(8,21,25,14,23,31,24,32)(18,28,36,34,35,33,22,30)Polycyclic presentation:
2^9|a^2=eg,b^2=ef,c^2=ef,d^2=fi,f^2=i,g^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=def,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386663);