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 | |
6724 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)(7,8)(9,17)(10,11)(12,14)(13,18)(15,19)(16,20)(22,26)(23,29)(28,33)(30,35);(1,2)(3,4)(6,11)(8,12)(13,17)(16,19)(22,26)(27,32)(28,33)(34,36);(6,11)(7,14)(13,17)(15,20)(21,22,25,26)(23,27,29,32)(24,28,31,33)(30,34,35,36);(5,7)(6,12)(8,11)(9,15)(10,14)(13,19)(16,17)(18,20)(21,23,25,29)(22,27,26,32)(24,30,31,35)(28,34,33,36);(5,8,10,12)(6,7,11,14)(9,16,18,19)(13,15,17,20)(21,24,25,31)(22,28,26,33)(23,30,29,35)(27,34,32,36);(5,9)(6,13)(7,15)(8,16)(10,18)(11,17)(12,19)(14,20)Polycyclic presentation:
2^9|c^2=i,d^2=i,e^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^b=ehi,d^c=dhi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7549305);