#8898258

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	#2404	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4252 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ghi,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8805454);