Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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)(5,6,9,7)(8,11,12,10)(13,14,17,19)(15,16,20,18)(21,22,26,27)(23,24,29,32)(25,28,35,36)(30,31,33,34);(8,12)(10,11)(13,14,17,19)(15,16,20,18)(22,27)(23,29)(28,36)(30,33);(6,7)(10,11)(13,15,17,20)(14,18,19,16)(21,23)(22,24)(25,33)(26,29)(27,32)(28,34)(30,35)(31,36);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15)(14,16)(17,20)(18,19)(21,24)(22,23)(25,34)(26,32)(27,29)(28,33)(30,36)(31,35);(5,8,9,12)(6,10,7,11)(13,16)(14,15)(17,18)(19,20)(21,25,26,35)(22,28,27,36)(23,30,29,33)(24,31,32,34)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,c^2=fh,d^2=fgh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,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, 4068456);