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 | #52004 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5170 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,6)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,20)(21,25)(22,23)(24,28)(26,27)(29,30)(31,32);(1,2)(3,7)(4,8)(5,6)(17,18,22,23)(19,20,24,28)(21,25,29,30)(26,27,31,32);(10,13)(14,15)(18,23)(19,24)(21,29)(27,32);(9,10)(11,14)(12,13)(15,16)(17,19)(18,20)(21,26)(22,24)(23,28)(25,27)(29,31)(30,32);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,16)(13,15)(17,20)(18,24)(19,23)(21,27)(22,28)(25,31)(26,30)(29,32);(1,4,5,3)(2,8,6,7)(17,21,22,29)(18,25,23,30)(19,26,24,31)(20,27,28,32)Polycyclic presentation:
2^9|b^2=i,e^2=gi,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7681153);