#8139480

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

4382 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8046676);