#8271085

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4151 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=i,d^2=ghi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8178281);