Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6356533 |
| 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 | |
1702 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(4,11)(7,12)(8,13)(10,15)(14,16)(18,22)(19,26)(20,23)(25,30)(27,28)(29,31);(1,2,4,6,5,8,11,13)(3,7,10,12,9,14,15,16)(17,18,23,30,21,25,20,22)(19,24,26,31,28,32,27,29);(17,19)(18,24)(20,27)(21,28)(22,29)(23,26)(25,32)(30,31);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16);(3,9)(7,14)(10,15)(12,16)Polycyclic presentation:
2^9|b^2=fi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,g^c=gi,e^d=ehi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72284038, #72338440, #72338441, … |
| maximal normal subgroup of | #72284038, #72338440, #72338441, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6263789);