#8270570

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

4300 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=i,d^2=gi,e^2=hi,f^2=i,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, 8177766);