Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #475185 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17029 |
| 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)(3,11)(4,8,16,24)(5,18)(6,10)(7,14)(9,19)(12,21)(13,22,29,27)(15,23)(17,26,20,25)(28,32,30,31);(1,3)(2,7,10,11)(4,13)(5,12)(6,14)(8,22,24,27)(9,21,18,23)(15,19)(16,29)(17,30)(20,28)(25,32,26,31);(3,12)(4,16)(5,19)(7,21)(8,24)(9,18)(11,23)(13,28)(14,15)(22,31)(27,32)(29,30);(1,4,5,20,6,16,19,17)(2,8,9,26,10,24,18,25)(3,13,12,28,14,29,15,30)(7,22,21,31,11,27,23,32)Polycyclic presentation:
2^9|a^2=ef,b^2=eh,d^2=eg,g^2=i,b^a=be,c^a=cf,d^a=dh,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, 382381);