Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
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 | |
6696 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 36 points:
(1,2,3,4)(7,13)(11,17)(14,19)(18,20)(21,22,27,29)(23,31,32,28)(24,34,33,30)(25,35,36,26);(1,2)(3,4)(5,6)(7,11)(8,15)(9,10)(12,16)(13,17)(14,20)(18,19)(21,23,27,32)(22,28,29,31)(24,25,33,36)(26,30,35,34);(6,10)(7,13)(12,15)(14,19)(22,29)(23,32)(25,36)(30,34);(5,7,9,13)(6,11,10,17)(8,14,16,19)(12,18,15,20)(21,24,27,33)(22,30,29,34)(23,25,32,36)(26,28,35,31);(5,6)(7,11)(8,12)(9,10)(13,17)(14,18)(15,16)(19,20)(21,25)(22,26)(23,33)(24,32)(27,36)(28,30)(29,35)(31,34);(5,8)(6,12)(7,14)(9,16)(10,15)(11,18)(13,19)(17,20)(21,26)(22,25)(23,34)(24,28)(27,35)(29,36)(30,32)(31,33)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7648462);