#8471785

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

4494 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8378981);