#8159148

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

4340 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=ghi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8066344);