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 | |
2046 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)(6,10)(7,12)(15,19)(18,20)(21,22)(23,27)(24,26)(25,32)(28,31)(29,34)(30,36)(33,35);(5,6)(7,11)(8,12)(9,10)(14,17)(18,20)(21,23,26,31)(22,27,24,28)(25,33,34,30)(29,36,32,35);(6,10)(8,11)(13,14,16,17)(15,18,19,20)(21,23)(22,28)(24,27)(25,30)(26,31)(29,36)(32,35)(33,34);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(13,15,16,19)(14,18,17,20)(21,24,26,22)(23,28,31,27)(25,32,34,29)(30,36,33,35);(5,8,9,11)(6,12,10,7)(21,25)(22,29)(23,30)(24,32)(26,34)(27,35)(28,36)(31,33)Polycyclic presentation:
2^9|b^2=g,c^2=fgh,d^2=fh,e^2=f,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, 4049429);