Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7339796 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #48088 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23875 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,7,10,18)(8,23,25,21)(9,22,28,14)(24,31,27,29);(1,2)(3,8)(4,14)(5,18)(6,10)(7,20)(9,26)(11,29)(12,21)(13,25)(15,31)(16,28)(17,22)(19,23)(24,32)(27,30);(1,3,6,13)(2,8,10,25)(4,15,17,11)(5,19,20,12)(7,21,18,23)(9,27,28,24)(14,29,22,31)(16,30,26,32);(1,4,5,16,6,17,20,26)(2,9,18,14,10,28,7,22)(3,11,12,30,13,15,19,32)(8,24,21,29,25,27,23,31);(1,2)(3,8)(4,9)(5,18)(6,10)(7,20)(11,24)(12,21)(13,25)(14,16)(15,27)(17,28)(19,23)(22,26)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=fg,f^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,h^d=hi,f^e=fi,g^e=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7246995);