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×#3566 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6424 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)(5,6,9,10)(7,11,13,17)(8,15,16,12)(14,20,19,18)(21,22,26,24)(23,29,30,27)(25,28,33,32)(31,35,36,34);(6,10)(7,13)(8,16)(18,20)(23,30)(25,33)(27,29)(28,32);(6,10)(7,13)(12,15)(14,19)(22,24)(27,29)(28,32)(34,35);(5,6)(7,11)(8,12)(9,10)(13,17)(14,18)(15,16)(19,20)(21,23,26,30)(22,27,24,29)(25,31,33,36)(28,34,32,35);(5,7,9,13)(6,11,10,17)(8,14,16,19)(12,18,15,20)(21,24,26,22)(23,29,30,27)(25,32,33,28)(31,35,36,34);(5,8,9,16)(6,12,10,15)(7,14,13,19)(11,18,17,20)(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=gh,e^2=h,f^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, 7534865);