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 | #2456 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10,13,11)(12,14,16,15)(17,18)(19,20)(21,29)(22,24)(23,26)(25,31)(27,30)(28,32);(1,3,5,8)(2,6,4,7)(17,19)(18,22)(20,26)(21,27)(23,24)(25,30)(28,29)(31,32);(10,11)(14,15)(17,20,23,22)(18,24,26,19)(21,30,31,28)(25,27,29,32);(9,11,13,10)(12,15,16,14)(17,21,23,31)(18,25,26,29)(19,27,24,32)(20,28,22,30);(1,2,5,4)(3,6,8,7)(9,12,13,16)(10,14,11,15)(17,19)(18,20)(21,27)(22,26)(23,24)(25,28)(29,30)(31,32);(1,4,5,2)(3,7,8,6)(17,22,23,20)(18,24,26,19)(21,30,31,28)(25,32,29,27)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=i,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^c=dh,f^e=fi107 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7606286);