#8696240

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

4339 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8603436);