Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #475397 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17046 |
| 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 | |
699 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,10,5,11,6,19)(3,12,16,23,15,8,13,21)(4,9,20,25,17,24,18,26)(14,22,28,32,29,27,30,31);(1,3)(2,8,11,12)(4,14)(5,15)(6,13)(7,16)(9,22,24,27)(10,21,19,23)(17,29)(18,30)(20,28)(25,32,26,31);(3,13)(4,17)(6,7)(8,21)(9,24)(10,19)(12,23)(14,28)(15,16)(22,31)(27,32)(29,30);(1,4,6,20,5,17,7,18)(2,9,10,26,11,24,19,25)(3,14,13,28,15,29,16,30)(8,22,21,31,12,27,23,32)Polycyclic presentation:
2^9|a^2=fg,b^2=eh,d^2=eg,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,g^c=gi,e^d=ei,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 382593);