#8346308

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

4517 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gi,d^2=hi,e^2=hi,f^2=hi,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, 8253504);