Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63041 |
| 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×#3508 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7182 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 36 points:
(1,2,3,4)(5,6)(7,15)(8,12)(9,13)(10,14)(11,16)(17,20)(18,19)(21,22)(23,30)(24,34)(25,27)(26,29)(28,33)(31,32)(35,36);(7,16)(8,17)(11,15)(12,20)(21,23)(22,28)(24,31)(25,30)(26,32)(27,33)(29,36)(34,35);(5,7,10,16)(6,11,14,15)(8,9,17,19)(12,18,20,13)(21,24)(22,26)(23,31)(25,34)(27,29)(28,32)(30,35)(33,36);(5,8,10,17)(6,12,14,20)(7,9,16,19)(11,18,15,13)(21,25,27,22)(23,28,33,30)(24,34,29,26)(31,32,36,35);(5,9)(6,13)(7,17)(8,16)(10,19)(11,12)(14,18)(15,20)(21,26,27,34)(22,29,25,24)(23,32,33,35)(28,36,30,31);(5,6)(7,11)(8,12)(9,18)(10,14)(13,19)(15,16)(17,20)(21,23)(22,30)(24,31)(25,28)(26,35)(27,33)(29,36)(32,34)Polycyclic presentation:
2^9|a^2=i,c^2=gh,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,d^b=dh,f^e=fh140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7534791);