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 | #36318 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4858 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10,13,11)(12,15,16,14)(17,18)(19,24)(20,22)(21,26)(23,25)(27,28)(29,32)(30,31);(1,2,5,6)(3,7,8,4)(10,11)(14,15)(17,19,23,26)(18,24,25,21)(20,27,29,31)(22,28,32,30);(2,6)(4,7)(9,11,13,10)(12,15,16,14)(18,25)(19,26)(22,32)(27,31);(2,6)(3,8)(9,12,13,16)(10,14,11,15)(17,20)(18,22)(19,27)(21,30)(23,29)(24,28)(25,32)(26,31);(1,3)(2,4)(5,8)(6,7)(17,21,23,24)(18,19,25,26)(20,28,29,30)(22,31,32,27);(1,4)(2,3)(5,7)(6,8)(17,22)(18,20)(19,28)(21,31)(23,32)(24,27)(25,29)(26,30)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=g,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8295362);