Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460350 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16362 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60454 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1222 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,8,13,7,11)(3,14,17,9,18,26,23,27)(4,19,5,10,22,28,21,29)(15,30,20,32,16,24,31,25);(1,3,6,17,8,18,7,23)(2,9,11,14,13,27,12,26)(4,20,5,16,22,31,21,15)(10,25,19,24,29,32,28,30);(1,4)(2,10)(3,15)(5,7)(6,21)(8,22)(9,24)(11,28)(12,19)(13,29)(14,25)(16,18)(17,31)(20,23)(26,32)(27,30);(1,5,7,4,8,21,6,22)(2,10,12,28,13,29,11,19)(3,16,23,20,18,15,17,31)(9,25,26,30,27,32,14,24)Polycyclic presentation:
2^9|a^2=g,b^2=e,d^2=ei,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^b=fi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72215936, #72216037, #72216047, … |
| maximal normal subgroup of | #72215936, #72216037, #72216047, … |
| maximal quotient of | C2×#460351, #72252212, #72255615, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367547);