#8273447

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

4446 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=g,e^2=gh,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, 8180643);