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 | |
6650 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 36 points:
(1,2,3,4)(8,16)(12,17)(14,20)(18,19)(21,22,26,28)(23,31,32,27)(24,29,34,36)(25,35,33,30);(1,2)(3,4)(5,6)(7,13)(8,12)(9,10)(11,15)(14,19)(16,17)(18,20)(21,23,26,32)(22,27,28,31)(24,25,34,33)(29,30,36,35);(6,10)(11,13)(12,17)(18,19)(22,28)(23,32)(24,34)(30,35);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,20)(17,19)(21,24,26,34)(22,29,28,36)(23,33,32,25)(27,35,31,30);(5,6)(7,11)(8,12)(9,10)(13,15)(14,18)(16,17)(19,20)(21,22)(23,27)(24,29)(25,30)(26,28)(31,32)(33,35)(34,36);(5,8,9,16)(6,12,10,17)(7,14,15,20)(11,18,13,19)(21,25,26,33)(22,30,28,35)(23,34,32,24)(27,36,31,29)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7622421);