Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6362510 |
| 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)(3,7)(4,11)(5,13)(6,9)(8,10)(12,15)(14,16)(17,18)(19,30)(20,31)(21,28)(22,27)(23,29)(24,26)(25,32);(1,3,6,5)(2,7,9,13)(4,10,12,14)(8,15,16,11)(17,19)(18,24)(20,29)(21,31)(22,26)(23,25)(27,32)(28,30);(1,4,6,12)(2,8,9,16)(3,10,5,14)(7,15,13,11)(17,20)(18,25)(19,29)(21,32)(22,30)(23,24)(26,28)(27,31);(17,20)(18,26)(19,27)(21,30)(22,32)(23,24)(25,28)(29,31);(2,9)(7,13)(8,16)(11,15)Polycyclic presentation:
2^9|b^2=hi,c^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ehi,f^a=fh,g^a=gi,d^b=df,f^b=fh,g^c=gi,f^d=fh71 conjugacy classes — view the full character table.
| maximal subgroup of | #72283366, #72283748, #72283969, … |
| maximal normal subgroup of | #72283366, #72283748, #72283969, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6269772);