Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460459 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16364 |
| 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 | |
947 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,7,17,20)(3,8,11,23,13,19,18,21)(4,14,15,29,16,9,24,26)(12,22,27,32,28,25,30,31);(2,7)(3,11)(5,17)(8,21)(9,14)(12,27)(13,18)(15,24)(19,23)(22,31)(25,32)(28,30);(1,3)(2,8,7,19)(4,12)(5,18)(6,13)(9,25,14,22)(10,21,20,23)(11,17)(15,30)(16,28)(24,27)(26,32,29,31);(1,4)(2,9,7,14)(3,12)(5,15)(6,16)(8,22,19,25)(10,26,20,29)(11,27)(13,28)(17,24)(18,30)(21,31,23,32)Polycyclic presentation:
2^9|a^2=g,c^2=e,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,d^c=de,f^c=fi,g^c=gi,h^c=hi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367655);