Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #474304 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16936 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
634 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,20,6,9,5,11)(3,12,14,23,15,26,21,8)(4,16,19,25,18,10,17,24)(13,27,28,22,29,32,30,31);(1,3,5,14,6,15,7,21)(2,8,20,26,9,23,11,12)(4,13,19,30,18,29,17,28)(10,22,25,32,16,31,24,27);(2,9)(4,17)(5,7)(10,24)(12,26)(13,28)(14,21)(16,25)(18,19)(22,32)(27,31)(29,30);(1,4)(2,10,9,16)(3,13)(5,19)(6,18)(7,17)(8,22,23,31)(11,24,20,25)(12,27,26,32)(14,30)(15,29)(21,28)Polycyclic presentation:
2^9|a^2=efg,b^2=efgh,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,e^d=ei,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 381500);