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 |
| prime radical | #2656 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6256 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)(9,17)(13,18)(14,19)(16,20)(21,22,26,28)(23,29,31,24)(25,33,34,27)(30,32,36,35);(1,2,3,4)(5,6)(7,8)(9,13)(10,11)(12,15)(14,16)(17,18)(19,20);(6,11)(8,15)(13,18)(16,20)(21,23,26,31)(22,24,28,29)(25,30,34,36)(27,32,33,35);(5,7,10,12)(6,8,11,15)(9,14,17,19)(13,16,18,20)(21,24)(22,23)(25,32)(26,29)(27,30)(28,31)(33,36)(34,35);(5,8)(6,12)(7,11)(9,16)(10,15)(13,19)(14,18)(17,20)(21,25,26,34)(22,27,28,33)(23,30,31,36)(24,32,29,35);(5,9)(6,13)(7,14)(8,16)(10,17)(11,18)(12,19)(15,20)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,d^2=i,e^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, 7624643);