#8429548

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	#52977	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4410 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,d^2=h,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^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, 8336744);