#8188119

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	#57732	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4371 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)(3,7)(4,5)(6,8)(9,10,14,12)(11,17,19,15)(13,16,21,20)(18,23,24,22)(25,26,32,31)(27,37,39,33)(28,40,38,34)(29,36,30,35);(1,2)(3,6)(4,5)(7,8)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,27)(26,33)(28,29)(30,38)(31,37)(32,39)(34,36)(35,40);(2,5)(6,7)(9,10,14,12)(11,15,19,17)(13,20,21,16)(18,23,24,22)(25,28,32,38)(26,34,31,40)(27,29,39,30)(33,36,37,35);(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24)(25,29,32,30)(26,35,31,36)(27,28,39,38)(33,40,37,34);(1,3)(2,6)(4,8)(5,7)(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,30,32,29)(26,36,31,35)(27,38,39,28)(33,34,37,40);(1,2)(3,6)(4,5)(7,8)(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(25,31,32,26)(27,37,39,33)(28,34,38,40)(29,36,30,35)

Polycyclic presentation:

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

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, 8095315);