#8288356

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

4399 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,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, 8195552);