Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6364250 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1615 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,9,11,15)(4,13,14,8)(6,10,16,12)(17,18,23,19)(20,28,30,32)(21,25,22,27)(24,31,26,29);(1,3,5,11)(2,6,7,16)(4,10,14,12)(8,15,13,9)(17,19,23,18)(20,26,30,24)(21,25,22,27)(28,29,32,31);(17,20)(18,24)(19,26)(21,29)(22,31)(23,30)(25,32)(27,28);(3,10)(4,14)(6,15)(8,13)(9,16)(11,12);(2,7)(3,11)(8,13)(10,12)(17,21,23,22)(18,25,19,27)(20,29,30,31)(24,32,26,28)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=ehi,f^b=fh,g^c=gi,f^d=fh65 conjugacy classes — view the full character table.
| maximal subgroup of | #72340114, #72340119, #72340120, … |
| maximal normal subgroup of | #72340114, #72340119, #72340120, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6271801);