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 | |
7039 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)(8,16)(11,18)(14,20)(17,19)(21,22)(23,33)(24,29)(25,30)(26,31)(27,32)(28,36)(34,35);(1,2)(3,4)(5,6)(7,12)(8,11)(9,13)(10,15)(14,19)(16,18)(17,20);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,23)(22,28)(24,34)(25,31)(26,35)(27,36)(29,30)(32,33);(5,8,9,16)(6,11,13,18)(7,14,15,20)(10,17,12,19)(21,24,27,31)(22,29,32,26)(23,34,36,25)(28,30,33,35);(5,7)(6,12)(8,14)(9,15)(10,13)(11,19)(16,20)(17,18)(21,25,27,34)(22,30,32,35)(23,31,36,24)(26,28,29,33);(5,6)(7,10)(8,11)(9,13)(12,15)(14,17)(16,18)(19,20)(21,26,27,29)(22,31,32,24)(23,35,36,30)(25,33,34,28)Polycyclic presentation:
2^9|d^2=hi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7564949);