#8429647

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4566 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8336843);