Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437986 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2410 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4505 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,11,7,12,6,10)(3,13,17,14,18,8,16,24)(4,9,20,19,21,26,22,25)(15,27,30,28,31,23,29,32);(1,2)(3,14)(4,19)(5,10)(6,11)(7,12)(8,16)(9,20)(13,17)(15,23)(18,24)(21,25)(22,26)(27,31)(28,30)(29,32);(1,3)(2,8)(4,15)(5,16)(6,17)(7,18)(9,23)(10,24)(11,14)(12,13)(19,28)(20,29)(21,31)(22,30)(25,32)(26,27);(1,4)(2,9)(3,15)(5,20)(6,22)(7,21)(8,23)(10,25)(11,19)(12,26)(13,27)(14,28)(16,29)(17,30)(18,31)(24,32);(1,5,7,6)(2,10,12,11)(3,16,18,17)(4,20,21,22)(8,24,13,14)(9,25,26,19)(15,29,31,30)(23,32,27,28)Polycyclic presentation:
2^9|a^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,h^b=hi,f^c=fi,h^c=hi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72215784, #72215931, #72219682, … |
| maximal normal subgroup of | #72215784, #72215931, #72219682, … |
| maximal quotient of | #72312228, C2×#6437988, #72316837, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345184);