Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7309140 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#127 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1691 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(17,18,22,24,20,25,21,26)(19,27,30,32,28,23,29,31);(2,5)(3,7)(10,14)(11,12)(17,19,20,28)(18,23,25,27)(21,29,22,30)(24,31,26,32);(9,10)(11,16)(12,15)(13,14)(18,24)(21,22)(23,31)(25,26)(27,32)(29,30);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16);(2,5)(6,8)(9,12,13,11)(10,15,14,16)(18,25)(23,27)(24,26)(31,32)Polycyclic presentation:
2^9|a^2=gi,b^2=i,d^2=fhi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7397069);