Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993887 |
| 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 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1620 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(18,24)(19,26)(20,28)(21,22)(25,29)(27,32);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,23)(20,29)(21,26)(22,24)(25,31)(27,30)(28,32);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,23,18)(20,25,28,27)(21,26,22,24)(29,30,32,31);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,20,23,28)(18,25,19,27)(21,30,22,31)(24,32,26,29);(2,8)(6,9)(7,16)(14,15)(17,21,23,22)(18,24,19,26)(20,30,28,31)(25,32,27,29)Polycyclic presentation:
2^9|c^2=h,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,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=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7031313);