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 | |
6786 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)(7,13)(8,12)(9,10)(11,15)(14,19)(16,17)(18,20)(21,22)(23,25)(24,32)(26,30)(27,29)(28,35)(31,36)(33,34);(1,2)(3,4)(21,23,26,29)(22,27,30,25)(24,33,35,31)(28,34,32,36);(6,10)(11,13)(12,17)(18,19)(21,22,26,30)(23,25,29,27)(24,32,35,28)(31,36,33,34);(5,6)(7,11)(8,12)(9,10)(13,15)(14,18)(16,17)(19,20)(21,22)(23,27)(24,28)(25,29)(26,30)(31,36)(32,35)(33,34);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,20)(17,19)(21,24,26,35)(22,28,30,32)(23,31,29,33)(25,34,27,36);(5,8,9,16)(6,12,10,17)(7,14,15,20)(11,18,13,19)(21,25,26,27)(22,29,30,23)(24,34,35,36)(28,33,32,31)Polycyclic presentation:
2^9|b^2=i,c^2=i,e^2=i,f^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7626281);