#8429372

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

4611 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=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, 8336568);