Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62972 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C26 |
| commutator | C22 | C25×C4 |
| prime radical | C2×#3565 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6230 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 36 points:
(1,2,3,4)(5,6)(7,13)(8,12)(9,10)(11,14)(15,19)(16,17)(18,20)(21,22,27,29)(23,32,34,28)(24,26,35,30)(25,31,33,36);(7,14)(8,16)(11,13)(12,17)(21,23,27,34)(22,28,29,32)(24,25,35,33)(26,36,30,31);(6,10)(11,13)(12,17)(18,19)(22,29)(24,35)(25,33)(28,32);(5,7)(6,11)(8,15)(9,14)(10,13)(12,18)(16,20)(17,19)(21,24)(22,30)(23,33)(25,34)(26,29)(27,35)(28,36)(31,32);(5,6)(7,11)(8,12)(9,10)(13,14)(15,18)(16,17)(19,20)(21,25,27,33)(22,31,29,36)(23,24,34,35)(26,28,30,32);(5,8,9,16)(6,12,10,17)(7,15,14,20)(11,18,13,19)(21,26,27,30)(22,24,29,35)(23,31,34,36)(25,28,33,32)Polycyclic presentation:
2^9|a^2=i,b^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fg,d^b=dh,e^b=eg,f^b=fh,d^c=dg,e^c=eh134 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7534868);