Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437950 |
| 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 | |
4533 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)(3,13,15,24,17,23,16,8)(4,9,19,18,20,26,21,25)(14,27,28,32,30,31,29,22);(1,2)(3,13)(4,18)(5,10)(6,11)(7,12)(8,15)(9,19)(14,22)(16,24)(17,23)(20,25)(21,26)(27,29)(28,31)(30,32);(1,3)(2,8)(4,14)(5,15)(6,16)(7,17)(9,22)(10,23)(11,13)(12,24)(18,27)(19,28)(20,30)(21,29)(25,31)(26,32);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,22)(10,25)(11,18)(12,26)(13,27)(15,28)(16,29)(17,30)(23,31)(24,32);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,20,21)(8,23,24,13)(9,25,26,18)(14,28,30,29)(22,31,32,27)Polycyclic presentation:
2^9|a^2=f,e^2=i,f^2=i,g^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,h^b=hi,g^c=gi,h^c=hi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72213913, #72214082, #72215721, … |
| maximal normal subgroup of | #72213913, #72214082, #72215721, … |
| maximal quotient of | #72250763, C2×#6437952, #72252147, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345148);