Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438013 |
| 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 | |
4608 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)(3,7)(4,15,17,8)(5,10)(6,9)(11,21,29,32)(12,22)(13,23)(14,24)(16,25,30,31)(18,28,19,26)(20,27);(1,3)(2,7)(4,16)(5,12)(6,13)(8,25)(9,23)(10,22)(11,18)(14,20)(15,31)(17,30)(19,29)(21,26)(24,27)(28,32);(1,4)(2,8)(3,11)(5,19)(6,17)(7,21)(9,15)(10,28)(12,16)(13,29)(14,30)(18,20)(22,25)(23,32)(24,31)(26,27);(3,12)(5,20)(7,22)(10,27)(11,16)(13,14)(18,19)(21,25)(23,24)(26,28)(29,30)(31,32);(2,9)(7,23)(8,15)(10,27)(21,32)(22,24)(25,31)(26,28)Polycyclic presentation:
2^9|a^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,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 | #72236873, #72237544, #72237551, … |
| maximal normal subgroup of | #72236873, #72237544, #72237551, … |
| maximal quotient of | #72237646, C2×#6438013, #72237694, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345209);