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 | |
6534 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)(5,6,11,13)(7,17,18,12)(8,10,16,14)(9,20,19,15)(21,22,26,29)(23,27,31,25)(24,32,34,28)(30,33,36,35);(1,2)(3,4)(5,7)(6,12)(8,19)(9,16)(10,20)(11,18)(13,17)(14,15);(6,13)(7,18)(8,16)(15,20)(21,23,26,31)(22,25,29,27)(24,30,34,36)(28,33,32,35);(5,8,11,16)(6,14,13,10)(7,19,18,9)(12,15,17,20)(21,23)(22,27)(24,30)(25,29)(26,31)(28,35)(32,33)(34,36);(5,9,11,19)(6,15,13,20)(7,16,18,8)(10,12,14,17)(21,24,26,34)(22,28,29,32)(23,30,31,36)(25,33,27,35);(5,10,11,14)(6,16,13,8)(7,20,18,15)(9,17,19,12)(21,25,26,27)(22,23,29,31)(24,33,34,35)(28,30,32,36)Polycyclic presentation:
2^9|a^2=h,c^2=hi,d^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7813069);