Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| 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 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1673 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(18,22)(23,31)(24,29)(25,32);(2,6)(7,8)(17,18,20,22)(19,25,26,32)(21,29,28,24)(23,30,31,27);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,19,20,26)(18,23,22,31)(21,30,28,27)(24,32,29,25);(1,4,5,3)(2,8,6,7)(18,24)(21,28)(22,29)(23,32)(25,31)(27,30);(9,11,12,15)(10,14,13,16)(18,22)(23,31)(24,29)(25,32)Polycyclic presentation:
2^9|b^2=i,c^2=f,d^2=fi,e^2=g,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,e^b=ei,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6363412);