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×#3519 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7162 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)(7,12)(8,11)(9,13)(10,15)(14,19)(16,17)(18,20)(21,22)(23,30)(24,34)(25,27)(26,29)(28,33)(31,32)(35,36);(5,6)(7,12)(8,16)(9,13)(10,15)(11,17)(14,18)(19,20)(21,23)(22,28)(24,31)(25,30)(26,32)(27,33)(29,36)(34,35);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,24)(22,26)(23,31)(25,34)(27,29)(28,32)(30,35)(33,36);(5,8)(6,11)(7,14)(9,17)(10,18)(12,19)(13,16)(15,20)(21,25,27,22)(23,28,33,30)(24,34,29,26)(31,32,36,35);(5,7)(6,12)(8,14)(9,15)(10,13)(11,19)(16,18)(17,20)(21,26,27,34)(22,29,25,24)(23,32,33,35)(28,36,30,31);(5,6)(7,10)(8,11)(9,13)(12,15)(14,18)(16,17)(19,20)(21,23)(22,30)(24,31)(25,28)(26,35)(27,33)(29,36)(32,34)Polycyclic presentation:
2^9|a^2=i,d^2=g,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, 7534654);