Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1938 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14,17,19)(15,20,18,16)(22,26)(23,29)(28,33)(30,35);(5,6,8,10)(7,11,12,9)(13,15,17,18)(14,16,19,20)(23,29)(27,32)(30,35)(34,36);(5,6,8,10)(7,9,12,11)(13,16,17,20)(14,18,19,15)(21,22,25,26)(23,27,29,32)(24,28,31,33)(30,34,35,36);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,16)(14,15)(17,20)(18,19)(21,23,25,29)(22,27,26,32)(24,30,31,35)(28,34,33,36);(5,7)(6,9)(8,12)(10,11)(13,15)(14,16)(17,18)(19,20)(21,24,25,31)(22,28,26,33)(23,30,29,35)(27,34,32,36)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=g,d^2=i,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4406958);