Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499874 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #35346 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3944 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)(4,8)(5,9)(6,10)(11,17)(12,18)(13,19)(14,21)(15,22)(16,24)(20,26)(23,28)(25,29)(27,31)(30,32);(1,3,6,13)(2,7,10,19)(4,11,15,26)(5,12,16,27)(8,20,22,17)(9,18,24,31)(14,25,28,32)(21,30,23,29);(1,4)(2,8)(3,11)(5,14)(6,15)(7,17)(9,23)(10,22)(12,25)(13,26)(16,28)(18,30)(19,20)(21,24)(27,32)(29,31);(1,5)(2,9)(3,12)(4,14)(6,16)(7,18)(8,21)(10,24)(11,25)(13,27)(15,28)(17,29)(19,31)(20,30)(22,23)(26,32);(1,4)(2,8)(3,11)(5,14)(6,15)(7,17)(9,21)(10,22)(12,25)(13,26)(16,28)(18,29)(19,20)(23,24)(27,32)(30,31)Polycyclic presentation:
2^9|b^2=i,b^a=bf,c^a=cg,h^a=hi,c^b=ch,e^b=eh,d^c=dh,f^c=fi,g^d=gi,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72218347, #72233445, #72233454, … |
| maximal normal subgroup of | #72218347, #72233445, #72233454, … |
| maximal quotient of | #72233435, #72234617, #72234643, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6407071);