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 | |
6640 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 36 points:
(2,4)(7,13)(11,14)(15,19)(18,20)(23,29)(25,33)(27,31)(28,32);(1,2)(3,4)(5,6)(7,14)(8,16)(9,10)(11,13)(12,17)(15,18)(19,20)(21,22,26,24)(23,27,29,31)(25,28,33,32)(30,34,35,36);(6,9)(11,14)(12,16)(18,20)(21,23)(22,27)(24,31)(25,30)(26,29)(28,34)(32,36)(33,35);(5,7)(6,11)(8,15)(9,14)(10,13)(12,18)(16,20)(17,19)(21,24,26,22)(23,27,29,31)(25,32,33,28)(30,34,35,36);(5,8)(6,12)(7,15)(9,16)(10,17)(11,18)(13,19)(14,20)(21,25,26,33)(22,28,24,32)(23,30,29,35)(27,34,31,36);(5,9,10,6)(7,14,13,11)(8,16,17,12)(15,20,19,18)Polycyclic presentation:
2^9|b^2=h,d^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=di,e^b=ei,f^b=fi,d^c=dh,f^c=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7718276);