Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1858 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(6,7)(8,12)(15,19)(18,20)(21,22,26,30)(23,31,33,27)(24,28,34,32)(25,29,35,36);(5,6,9,7)(8,10,12,11)(14,17)(18,20)(21,23,26,33)(22,27,30,31)(24,29,34,36)(25,32,35,28);(5,6,9,7)(8,11,12,10)(13,14)(15,18)(16,17)(19,20)(21,24)(22,28)(23,29)(25,27)(26,34)(30,32)(31,35)(33,36);(1,2,3,4)(5,7,9,6)(8,10,12,11)(13,15)(14,18)(16,19)(17,20)(21,23)(22,27)(24,29)(25,28)(26,33)(30,31)(32,35)(34,36);(5,8,9,12)(6,10,7,11)(21,25)(22,29)(23,32)(24,31)(26,35)(27,34)(28,33)(30,36)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=g,d^2=fghi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3376148);