#8760689

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

4373 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8667885);