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 | |
3873 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)(3,11)(4,8)(5,17)(6,10)(7,13)(9,19)(12,25)(14,24)(15,28)(16,22)(18,20)(21,26)(23,30)(27,32)(29,31);(1,3,6,14)(2,7,10,20)(4,12,16,27)(5,18,19,13)(8,21,22,31)(9,24,17,11)(15,29,30,26)(23,32,28,25);(7,20)(8,22)(11,24)(23,28);(1,4)(2,8)(3,12)(5,15)(6,16)(7,21)(9,23)(10,22)(11,25)(13,26)(14,27)(17,28)(18,29)(19,30)(20,31)(24,32);(5,19)(7,20)(9,17)(13,18)(15,30)(21,31)(23,28)(26,29)Polycyclic presentation:
2^9|b^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 | #72234419, #72251420, #72251567, … |
| maximal normal subgroup of | #72234419, #72251420, #72251567, … |
| maximal quotient of | #72233091, #72256141, #72256149, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6407052);