Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420487 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15278 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1639 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)(3,13)(4,7)(5,20,23,10)(6,11)(8,12)(9,16)(14,18)(15,27,30,29)(17,26)(19,24,28,22)(21,25,31,32);(1,3)(2,9)(4,14)(5,21)(6,17)(7,16)(8,18)(10,27)(11,13)(12,26)(15,22)(19,25)(20,29)(23,31)(24,30)(28,32);(1,4,6,12,8,11,7,2)(3,14,16,9,18,13,17,26)(5,19,22,20,23,28,24,10)(15,29,21,25,30,27,31,32);(1,5,6,22,8,23,7,24)(2,10,4,19,12,20,11,28)(3,15,16,21,18,30,17,31)(9,25,13,27,26,32,14,29)Polycyclic presentation:
2^9|a^2=ef,c^2=e,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,g^c=gi,h^c=hi,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327683);