#8701305

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×#127	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4271 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=ghi,e^2=gh,f^2=gi,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, 8608501);