#8416891

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	#52480	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4229 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=g,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8324087);