#8197272

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

4457 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)(3,7)(4,5)(6,8)(9,10,14,15)(11,12,17,16)(13,18,21,22)(19,20,24,23)(25,26,31,33)(27,34,28,32)(29,39,40,30)(35,36,37,38);(1,2)(3,6)(4,5)(7,8)(10,15)(11,17)(18,22)(19,24)(25,27,31,28)(26,32,33,34)(29,35,40,37)(30,36,39,38);(2,5)(6,7)(13,21)(18,22)(19,24)(20,23)(26,33)(30,39)(32,34)(36,38);(9,11,14,17)(10,16,15,12)(13,19,21,24)(18,23,22,20)(25,28,31,27)(26,34,33,32)(29,37,40,35)(30,38,39,36);(1,3)(2,6)(4,8)(5,7)(9,12,14,16)(10,17,15,11)(13,20,21,23)(18,24,22,19)(25,29)(26,30)(27,35)(28,37)(31,40)(32,36)(33,39)(34,38);(1,2)(3,6)(4,5)(7,8)(9,13,14,21)(10,18,15,22)(11,19,17,24)(12,20,16,23)(25,30)(26,29)(27,36)(28,38)(31,39)(32,35)(33,40)(34,37)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=g,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8104468);