#8126970

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	#51630	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4250 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8034166);