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 | |
7000 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)(5,6)(7,14)(8,16)(9,12)(10,13)(11,18)(15,17)(19,20)(21,22,26,24)(23,27,29,30)(25,28,33,32)(31,34,35,36);(1,2)(3,4)(23,29)(27,30)(31,35)(34,36);(5,7,10,12)(6,9,13,14)(8,15,18,19)(11,17,16,20)(21,23,26,29)(22,27,24,30)(25,31,33,35)(28,34,32,36);(5,8)(6,11)(7,15)(9,17)(10,18)(12,19)(13,16)(14,20);(5,6)(7,9)(8,11)(10,13)(12,14)(15,17)(16,18)(19,20)(21,24,26,22)(23,30,29,27)(25,28,33,32)(31,34,35,36);(5,9)(6,12)(7,13)(8,17)(10,14)(11,19)(15,16)(18,20)(21,25,26,33)(22,28,24,32)(23,31,29,35)(27,34,30,36)Polycyclic presentation:
2^9|a^2=gi,c^2=hi,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7562128);