#9062557

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4297 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(9,10,15,20)(11,21,24,16)(12,14,19,17)(13,18,22,23)(25,26,30,35)(27,31,38,32)(28,33,34,29)(36,37,39,40);(1,2)(3,7)(4,5)(6,8)(9,11,15,24)(10,16,20,21)(12,13,19,22)(14,23,17,18)(25,27)(26,31)(28,39)(29,37)(30,38)(32,35)(33,40)(34,36);(2,5)(3,8)(12,19)(13,22)(14,17)(18,23)(25,27,30,38)(26,32,35,31)(28,39,34,36)(29,40,33,37);(1,3,4,8)(2,6,5,7)(9,12,15,19)(10,17,20,14)(11,22,24,13)(16,23,21,18)(25,28,30,34)(26,33,35,29)(27,36,38,39)(31,37,32,40);(1,3)(2,6)(4,8)(5,7)(9,13,15,22)(10,18,20,23)(11,12,24,19)(14,16,17,21)(25,27)(26,31)(28,39)(29,37)(30,38)(32,35)(33,40)(34,36);(1,2)(3,6)(4,5)(7,8)(9,14,15,17)(10,19,20,12)(11,23,24,18)(13,16,22,21)(25,29,30,33)(26,34,35,28)(27,37,38,40)(31,36,32,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=gi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8969753);