#8346170

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4693 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gi,d^2=gi,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=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, 8253366);