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 | |
1712 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,8,10)(7,9,11,12)(14,17)(18,20)(21,22,27,30)(23,29,26,28)(24,33,34,25)(31,32,35,36);(2,4)(15,19)(18,20)(21,23,27,26)(22,28,30,29)(24,31,34,35)(25,32,33,36);(1,2,3,4)(13,14)(15,18)(16,17)(19,20)(21,24)(22,25)(23,31)(26,35)(27,34)(28,32)(29,36)(30,33);(5,6)(7,9)(8,10)(11,12)(13,15,16,19)(14,18,17,20)(21,25)(22,24)(23,32)(26,36)(27,33)(28,31)(29,35)(30,34);(5,7,8,11)(6,9,10,12)(13,14)(15,18)(16,17)(19,20)(21,26,27,23)(22,29,30,28)(24,31,34,35)(25,32,33,36)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=h,d^2=fg,e^2=fi,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, 3540228);