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 | |
6942 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 36 points:
(2,4)(5,6,10,12)(7,15,16,11)(8,19,18,13)(9,14,17,20)(21,22)(23,27)(24,32)(25,29)(26,30)(28,34)(31,36)(33,35);(1,2)(3,4)(5,7,10,16)(6,11,12,15)(8,17,18,9)(13,20,19,14)(21,23,26,25)(22,27,30,29)(24,31,34,33)(28,35,32,36);(6,12)(7,16)(8,18)(14,20)(21,22,26,30)(23,27,25,29)(24,28,34,32)(31,35,33,36);(5,6)(7,11)(8,13)(9,14)(10,12)(15,16)(17,20)(18,19)(21,22)(23,27)(24,28)(25,29)(26,30)(31,35)(32,34)(33,36);(5,8)(6,13)(7,17)(9,16)(10,18)(11,20)(12,19)(14,15)(21,24)(22,28)(23,31)(25,33)(26,34)(27,35)(29,36)(30,32);(5,9,10,17)(6,14,12,20)(7,18,16,8)(11,19,15,13)(21,25,26,23)(22,29,30,27)(24,33,34,31)(28,36,32,35)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=hi,f^2=h,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, 7882218);