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×#3537 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7250 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)(7,10)(8,11)(9,14)(12,17)(13,15)(16,19)(18,20)(21,22,26,24)(23,27,29,30)(25,32,33,28)(31,36,35,34);(6,7)(8,15)(9,17)(18,19)(22,24)(27,30)(28,32)(34,36);(6,7)(8,15)(12,14)(16,20)(23,29)(27,30)(31,35)(34,36);(5,7,10,6)(8,13,15,11)(9,14,17,12)(16,19,20,18)(21,23,26,29)(22,27,24,30)(25,31,33,35)(28,34,32,36);(5,8,10,15)(6,11,7,13)(9,16,17,20)(12,18,14,19)(21,24,26,22)(23,30,29,27)(25,32,33,28)(31,36,35,34);(5,9)(6,12)(7,14)(8,16)(10,17)(11,18)(13,19)(15,20)(21,25)(22,28)(23,31)(24,32)(26,33)(27,34)(29,35)(30,36)Polycyclic presentation:
2^9|a^2=i,d^2=g,e^2=g,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, 7534821);