Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #368558 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14020 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1207 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,11,6,10,13,14)(3,12,17,26,16,4,8,19)(5,20,21,27,22,9,23,25)(15,18,24,31,29,28,30,32);(1,3,13,17,6,16,7,8)(2,4,14,19,10,12,11,26)(5,15,21,30,22,29,23,24)(9,18,25,32,20,28,27,31);(1,4)(2,8,10,17)(3,14,16,11)(5,18)(6,12)(7,19)(9,24,20,30)(13,26)(15,25,29,27)(21,31)(22,28)(23,32);(1,5)(2,9)(3,15)(4,18)(6,22)(7,23)(8,24)(10,20)(11,27)(12,28)(13,21)(14,25)(16,29)(17,30)(19,32)(26,31)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=eg,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,f^d=fi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72319444, #72319453, #72319471, … |
| maximal normal subgroup of | #72319444, #72319453, #72319471, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275755);