Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
2136 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)(15,19)(16,18)(17,20)(21,22,26,30)(23,27,28,24)(25,29,34,36)(31,35,33,32);(7,11)(9,12)(13,14)(15,19)(16,18)(17,20)(21,23,26,28)(22,24,30,27)(25,31,34,33)(29,32,36,35);(5,6,8,10)(7,9,11,12)(13,14,16,18)(15,17,20,19)(21,23)(22,27)(24,30)(25,33)(26,28)(29,32)(31,34)(35,36);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,17)(16,18)(19,20)(21,24,26,27)(22,28,30,23)(25,32,34,35)(29,33,36,31);(5,7)(6,9)(8,11)(10,12)(13,15,16,20)(14,17,18,19)(21,25)(22,29)(23,31)(24,32)(26,34)(27,35)(28,33)(30,36)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=g,d^2=i,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5188590);