Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499855 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #35346 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3812 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)(3,10)(4,9)(5,11)(6,20)(7,13)(8,16)(12,22)(14,26)(15,28)(17,29)(18,30)(19,27)(21,25)(23,32)(24,31);(1,3)(2,8)(4,14)(5,15)(6,21)(7,17)(9,24)(10,20)(11,23)(12,29)(13,25)(16,22)(18,32)(19,31)(26,30)(27,28);(1,4,7,5)(2,9,13,11)(3,14,17,15)(6,18,22,19)(8,23,25,24)(10,28,29,26)(12,27,20,30)(16,31,21,32);(1,3)(2,10)(4,14)(5,15)(6,16)(7,17)(8,20)(9,26)(11,28)(12,25)(13,29)(18,31)(19,32)(21,22)(23,27)(24,30);(1,5,7,4)(2,11,13,9)(3,15,17,14)(6,18,22,19)(8,24,25,23)(10,28,29,26)(12,27,20,30)(16,31,21,32)Polycyclic presentation:
2^9|c^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,g^d=gi,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72261130, #72261154, #72306382, … |
| maximal normal subgroup of | #72261130, #72261154, #72306382, … |
| maximal quotient of | #72267104, #72286224, #72306371, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6407056);