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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5161 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10,13,14)(11,12,15,16)(17,18)(19,26)(20,21)(22,24)(23,29)(25,31)(27,28)(30,32);(1,2,5,6)(3,7,8,4)(10,14)(11,15)(17,19,22,29)(18,23,24,26)(20,30,31,27)(21,28,25,32);(2,6)(4,7)(9,11)(10,12)(13,15)(14,16)(18,24)(21,25)(23,26)(28,32);(1,3,5,8)(2,7,6,4)(9,12)(10,11)(13,16)(14,15)(17,19)(18,23)(20,30)(21,28)(22,29)(24,26)(25,32)(27,31);(1,3)(2,4)(5,8)(6,7)(17,20)(18,25)(19,27)(21,24)(22,31)(23,28)(26,32)(29,30);(1,4)(2,3)(5,7)(6,8)(17,21,22,25)(18,20,24,31)(19,28,29,32)(23,30,26,27)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,d^2=ghi,f^2=h,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, 8266362);