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 | |
4560 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10,13,11)(12,15,16,14)(17,18)(19,25)(20,22)(21,28)(23,26)(24,29)(27,30)(31,32);(1,2,5,4)(3,7,8,6)(10,11)(14,15)(18,20)(19,26)(24,28)(27,32);(2,4)(6,7)(9,11,13,10)(12,15,16,14)(17,19)(18,23)(20,25)(21,27)(22,26)(24,30)(28,31)(29,32);(3,8)(6,7)(9,12,13,16)(10,14,11,15)(17,20,22,18)(19,25,26,23)(21,28,29,24)(27,31,32,30);(1,3)(2,6)(4,7)(5,8)(17,21,22,29)(18,24,20,28)(19,27,26,32)(23,30,25,31);(1,4,5,2)(3,7,8,6)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=gh,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8282005);