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 | |
4916 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)(10,12)(14,15)(17,18)(19,22)(20,25)(21,24)(23,27)(26,30)(28,32)(29,31);(1,2,5,6)(3,7,4,8)(17,18,22,19)(20,23,27,25)(21,26,30,24)(28,32,29,31);(1,3,5,4)(2,7,6,8)(9,10,13,12)(11,15,16,14)(18,19)(20,27)(24,26)(28,29);(9,11,13,16)(10,14,12,15)(17,19,22,18)(20,25,27,23)(21,26,30,24)(28,32,29,31);(9,12,13,10)(11,15,16,14)(17,20)(18,23)(19,25)(21,29)(22,27)(24,32)(26,31)(28,30);(1,4,5,3)(2,8,6,7)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)Polycyclic presentation:
2^9|b^2=g,c^2=gh,d^2=h,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7895413);