Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6356901 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1662 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,7,9)(3,12,13,16)(4,10,8,15)(5,11,6,14)(17,18)(19,21)(20,27)(22,25)(23,26)(24,30)(28,31)(29,32);(1,3,5,15,7,13,6,10)(2,8,14,16,9,4,11,12)(17,19,21,18,23,25,22,26)(20,28,29,32,31,27,30,24);(2,9)(3,13)(10,15)(11,14)(17,20,23,31)(18,24,26,32)(19,27,25,28)(21,29,22,30);(1,4)(2,10)(3,14)(5,12)(6,16)(7,8)(9,15)(11,13)(17,21,23,22)(18,19,26,25)(20,29,31,30)(24,27,32,28);(1,5,7,6)(2,11,9,14)(3,15,13,10)(4,16,8,12)Polycyclic presentation:
2^9|a^2=hi,b^2=fh,c^2=i,d^2=i,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,d^b=dh,g^c=gi,e^d=ehi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72338303, #72338415, #72338416, … |
| maximal normal subgroup of | #72338303, #72338415, #72338416, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264184);