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 | |
6488 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)(7,15)(8,10)(9,12)(11,14)(13,20)(16,17)(18,19)(21,22,26,30)(23,31,32,27)(24,28,25,29)(33,36,34,35);(1,2,3,4)(5,7,11,9)(6,12,14,15)(8,18,13,16)(10,17,20,19)(23,32)(27,31)(33,34)(35,36);(7,9)(12,15)(16,18)(17,19)(21,23)(22,27)(24,33)(25,34)(26,32)(28,35)(29,36)(30,31);(5,8,11,13)(6,10,14,20)(7,16,9,18)(12,19,15,17)(21,22)(23,27)(24,28)(25,29)(26,30)(31,32)(33,35)(34,36);(5,9,11,7)(6,12,14,15)(8,18,13,16)(10,19,20,17)(21,24,26,25)(22,28,30,29)(23,33,32,34)(27,35,31,36);(5,10)(6,13)(7,17)(8,14)(9,19)(11,20)(12,16)(15,18)(21,25,26,24)(22,29,30,28)(23,34,32,33)(27,36,31,35)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=hi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,d^b=dhi,e^c=ehi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7551539);