Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993875 |
| 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 | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1624 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,6,9,13)(4,11,12,7)(10,16,15,14)(18,24)(19,26)(21,22)(25,32)(27,28)(29,31);(3,9)(4,12)(6,13)(7,11)(17,18,23,19)(20,28,30,32)(21,26,22,24)(25,31,27,29);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,19,23,18)(20,27,30,25)(21,26,22,24)(28,29,32,31);(3,9)(6,13)(10,15)(14,16)(17,20)(18,25)(19,27)(21,31)(22,29)(23,30)(24,32)(26,28);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7073033);