Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417442 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#1999 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1838 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,3,4)(9,10,13,12)(11,15,16,14)(17,18,20,25,22,26,21,24)(19,23,27,31,29,32,28,30);(5,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,19,22,29)(18,23,26,32)(20,27,21,28)(24,30,25,31);(2,4)(6,8)(9,12,13,10)(11,15,16,14)(18,24)(20,21)(23,30)(25,26)(27,28)(31,32);(11,16)(14,15)(18,25)(20,21)(23,31)(24,26)(27,28)(30,32)Polycyclic presentation:
2^9|a^2=ef,b^2=i,c^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,f^c=fi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326617);