#8324871

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

4331 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=h,d^2=g,e^2=g,f^2=gi,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, 8232067);