#8159935

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

4597 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,17)(13,18)(15,20)(16,21)(19,22)(23,24)(25,26)(27,31)(28,36)(29,37)(30,32)(33,39)(34,40)(35,38);(1,3,5,4)(2,6,7,8)(10,11)(15,17)(16,18)(22,23)(25,27,30,35)(26,31,32,38)(28,34,39,37)(29,36,40,33);(1,2)(3,6)(4,8)(5,7)(10,11)(15,17)(16,18)(22,23)(26,32)(27,35)(33,36)(34,37);(2,7)(6,8)(9,11,14,10)(12,15,20,17)(13,18,21,16)(19,22,24,23)(25,27)(26,31)(28,37)(29,36)(30,35)(32,38)(33,40)(34,39);(1,2)(3,6)(4,8)(5,7)(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,28,30,39)(26,33,32,36)(27,34,35,37)(29,38,40,31);(1,4,5,3)(2,8,7,6)(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,29)(26,34)(27,36)(28,38)(30,40)(31,39)(32,37)(33,35)

Polycyclic presentation:

2^9|b^2=gh,d^2=hi,e^2=h,f^2=g,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, 8067131);