Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #458575 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16320 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1062 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)(3,7,12,21)(4,8)(5,17)(6,9)(10,19)(11,27,28,20)(13,26,18,22)(14,23,29,32)(15,30)(16,24)(25,31);(1,3)(2,7,9,21)(4,14)(5,18)(6,12)(8,23,24,32)(10,26,17,22)(11,15)(13,19)(16,29)(20,30,27,25)(28,31);(1,4)(2,8)(3,11,12,28)(5,15)(6,16)(7,20,21,27)(9,24)(10,25)(13,14,18,29)(17,30)(19,31)(22,23,26,32);(5,19)(8,24)(10,17)(11,28)(13,18)(15,31)(22,26)(23,32)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,h^c=hi,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72319831, #72320054, #72320088, … |
| maximal normal subgroup of | #72319831, #72320054, #72320088, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 365771);