#8621955

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

4432 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)(4,6)(5,8)(9,10,14,15)(11,19,20,16)(12,13,18,17)(21,22,24,23)(25,26)(27,33)(28,30)(29,32)(31,35)(34,40)(36,38)(37,39);(1,3,6,8)(2,7,4,5)(10,15)(11,20)(12,18)(22,23)(25,27)(26,31)(28,33)(29,37)(30,35)(32,40)(34,38)(36,39);(2,4)(3,8)(10,15)(12,18)(16,19)(21,24)(29,38)(32,36)(34,37)(39,40);(3,8)(5,7)(9,11,14,20)(10,16,15,19)(12,21,18,24)(13,23,17,22)(25,28,30,26)(27,33,35,31)(29,36,38,32)(34,40,37,39);(1,4,6,2)(3,7,8,5)(9,12,14,18)(10,17,15,13)(11,21,20,24)(16,22,19,23)(25,29,30,38)(26,32,28,36)(27,34,35,37)(31,39,33,40);(1,5,6,7)(2,8,4,3)(9,13,14,17)(10,18,15,12)(11,22,20,23)(16,21,19,24)(25,27)(26,33)(28,31)(29,34)(30,35)(32,40)(36,39)(37,38)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8529151);