Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62862 |
| 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×#3553 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 36 points:
(1,2,4,3)(5,6)(7,11)(8,9)(10,12)(13,17)(14,15)(16,18)(19,20)(21,22,26,29)(23,24,31,27)(25,28,33,35)(30,32,36,34);(5,7,10,16)(6,11,12,18)(8,14,13,19)(9,15,17,20)(21,23,26,31)(22,27,29,24)(25,30,33,36)(28,34,35,32);(6,12)(8,13)(11,18)(14,19)(21,23)(22,24)(25,30)(26,31)(27,29)(28,32)(33,36)(34,35);(5,8,10,13)(6,9,12,17)(7,14,16,19)(11,15,18,20)(21,24)(22,23)(25,32)(26,27)(28,30)(29,31)(33,34)(35,36);(1,3,4,2)(5,7)(6,11)(8,14)(9,15)(10,16)(12,18)(13,19)(17,20);(5,9)(6,13)(7,15)(8,12)(10,17)(11,19)(14,18)(16,20)(21,25,26,33)(22,28,29,35)(23,30,31,36)(24,32,27,34)Polycyclic presentation:
2^9|a^2=i,b^2=gh,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh,d^c=dh140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533704);