#8429483

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, prime radical, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4109 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=hi,d^2=g,e^2=h,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, 8336679);