Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #421447 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2474 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15772 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
810 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)(3,13,17,23)(4,19,22,27)(5,6)(7,11)(8,12)(9,16,15,18)(10,21,28,24)(14,25)(20,26)(29,30)(31,32);(1,3,7,16,8,17,6,18)(2,9,11,13,12,15,5,23)(4,20,21,30,22,14,24,31)(10,26,19,29,28,25,27,32);(1,4,6,21,8,22,7,24)(2,10,5,19,12,28,11,27)(3,14,18,31,17,20,16,30)(9,25,23,32,15,26,13,29);(1,5,6,12,8,11,7,2)(3,15,16,23,17,9,18,13)(4,19,21,28,22,27,24,10)(14,26,30,32,20,25,31,29)Polycyclic presentation:
2^9|a^2=ef,b^2=f,c^2=eg,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,f^c=fi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72319815 |
| maximal normal subgroup of | #72319815 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 351281);