#8269876

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

4553 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:

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

Polycyclic presentation:

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

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, 8177072);