Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #45064 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4633 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,6,4)(3,7,5,8)(9,10,13,12)(11,15,16,14)(17,18,22,24)(19,26,20,23)(21,29,30,25)(27,31,28,32);(1,2)(3,8)(4,6)(5,7)(9,11,13,16)(10,14,12,15)(17,19,22,20)(18,23,24,26)(21,27,30,28)(25,31,29,32);(1,3,6,5)(2,7,4,8)(18,24)(19,20)(25,29)(27,28);(17,20,22,19)(18,23,24,26)(21,28,30,27)(25,31,29,32);(1,4,6,2)(3,7,5,8)(17,18)(19,26)(20,23)(21,25)(22,24)(27,32)(28,31)(29,30);(1,5,6,3)(2,7,4,8)(9,12,13,10)(11,15,16,14)(17,21,22,30)(18,25,24,29)(19,27,20,28)(23,31,26,32)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=hi,d^2=h,e^2=hi,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7626272);