#8696071

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

4608 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gi,d^2=g,e^2=h,f^2=g,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, 8603267);