#8345913

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

4444 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)(3,8)(4,6)(5,7)(9,10,15,12)(11,16,14,18)(13,17,22,20)(19,23,21,24)(25,26,31,30)(27,32,34,28)(29,37,39,33)(35,36,40,38);(1,2,6,4)(3,8,5,7)(9,11,15,14)(10,16,12,18)(13,19,22,21)(17,23,20,24)(25,27)(26,32)(28,30)(29,35)(31,34)(33,38)(36,37)(39,40);(1,2,6,4)(3,7,5,8)(11,14)(16,18)(19,21)(23,24)(26,30)(27,34)(33,37)(35,40);(1,3,6,5)(2,7,4,8)(9,12,15,10)(11,18,14,16)(13,17,22,20)(19,23,21,24)(25,28)(26,27)(29,38)(30,34)(31,32)(33,40)(35,37)(36,39);(1,4,6,2)(3,7,5,8)(9,13,15,22)(10,17,12,20)(11,19,14,21)(16,23,18,24)(25,29)(26,33)(27,35)(28,36)(30,37)(31,39)(32,38)(34,40);(1,5,6,3)(2,7,4,8)(9,14,15,11)(10,18,12,16)(13,21,22,19)(17,24,20,23)(25,30,31,26)(27,28,34,32)(29,37,39,33)(35,36,40,38)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=g,d^2=hi,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=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, 8253109);