Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478779 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17167 |
| 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 | |
661 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,21,6,10,5,11)(3,8,14,23,15,24,13,27)(4,16,20,9,18,28,19,26)(12,29,31,25,17,32,30,22);(1,3)(2,8)(4,17,18,12)(5,13)(6,15)(7,14)(9,25,26,22)(10,24)(11,27)(16,29,28,32)(19,31,20,30)(21,23);(1,4,5,20,6,18,7,19)(2,9,11,28,10,26,21,16)(3,12,14,30,15,17,13,31)(8,22,23,32,24,25,27,29);(2,10)(3,13)(4,18)(5,7)(8,23)(12,30)(14,15)(16,28)(17,31)(22,29)(24,27)(25,32)Polycyclic presentation:
2^9|a^2=efg,b^2=e,c^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=def,f^b=fi,g^b=gi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385975);