Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62892 |
| 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×#3514 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6956 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 36 points:
(1,2,4,3)(5,6,9,13)(7,11,15,10)(8,16,17,12)(14,18,20,19)(21,22)(23,30)(24,27)(25,33)(26,29)(28,34)(31,32)(35,36);(1,3,4,2)(5,7)(6,10)(8,14)(9,15)(11,13)(12,18)(16,19)(17,20);(5,6)(7,11)(8,12)(9,13)(10,15)(14,19)(16,17)(18,20)(21,23,26,27)(22,24,29,30)(25,31,34,35)(28,32,33,36);(5,6)(7,10)(8,12)(9,13)(11,15)(14,18)(16,17)(19,20)(21,22)(23,24)(25,28)(26,29)(27,30)(31,32)(33,34)(35,36);(5,7)(6,11)(8,14)(9,15)(10,13)(12,19)(16,18)(17,20)(21,24)(22,27)(23,29)(25,32)(26,30)(28,35)(31,33)(34,36);(5,8,9,17)(6,12,13,16)(7,14,15,20)(10,18,11,19)(21,25)(22,28)(23,31)(24,32)(26,34)(27,35)(29,33)(30,36)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^d=eh146 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533882);