#8739281

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	#58380	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4182 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)(3,5)(4,7)(6,8)(10,12)(15,18)(16,20)(22,24)(25,26,31,36)(27,34,29,32)(28,33,39,38)(30,35,40,37);(2,3)(6,7)(11,17)(15,18)(19,23)(22,24)(25,27,31,29)(26,32,36,34)(28,35,39,37)(30,38,40,33);(1,2,5,3)(4,6,8,7)(9,10,14,12)(11,15,17,18)(13,16,21,20)(19,22,23,24)(25,27,31,29)(26,32,36,34)(28,35,39,37)(30,38,40,33);(1,3,5,2)(4,7,8,6)(9,11,14,17)(10,15,12,18)(13,19,21,23)(16,22,20,24)(25,28,31,39)(26,33,36,38)(27,37,29,35)(30,32,40,34);(1,4,5,8)(2,6,3,7)(9,12,14,10)(11,18,17,15)(13,20,21,16)(19,24,23,22)(25,29,31,27)(26,34,36,32)(28,35,39,37)(30,38,40,33);(9,13,14,21)(10,16,12,20)(11,19,17,23)(15,22,18,24)(25,30)(26,35)(27,38)(28,34)(29,33)(31,40)(32,39)(36,37)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=h,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8646477);