Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #432847 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15467 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1593 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,5,12,7,9)(3,13,16,24,17,8,14,23)(4,18,21,27,19,25,20,10)(15,29,30,22,31,32,28,26);(1,3,7,16,5,17,6,14)(2,8,11,24,12,13,9,23)(4,15,21,28,19,31,20,30)(10,26,25,29,27,22,18,32);(2,9)(3,14)(6,7)(8,13)(10,25)(11,12)(15,28)(16,17)(18,27)(20,21)(22,26)(30,31);(1,4)(2,10)(3,15)(5,19)(6,21)(7,20)(8,22)(9,25)(11,18)(12,27)(13,26)(14,28)(16,30)(17,31)(23,32)(24,29)Polycyclic presentation:
2^9|a^2=eg,b^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72237070, #72237071, #72237090, … |
| maximal normal subgroup of | #72237070, #72237071, #72237090, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 340043);