#8696725

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

4656 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)(4,6)(5,8)(9,10,15,16)(11,13,12,17)(14,18,22,23)(19,21,20,24)(25,26)(27,32)(28,34)(29,38)(30,36)(31,37)(33,39)(35,40);(1,2,4,6)(3,5,8,7)(10,16)(11,12)(18,23)(19,20)(25,27,31,30)(26,32,37,36)(28,35,39,38)(29,34,40,33);(1,2)(3,5)(4,6)(7,8)(9,11,15,12)(10,17,16,13)(14,19,22,20)(18,24,23,21)(25,28)(26,33)(27,38)(29,36)(30,35)(31,39)(32,40)(34,37);(1,3,4,8)(2,5,6,7)(9,12,15,11)(10,13,16,17)(14,20,22,19)(18,21,23,24)(25,28)(26,34)(27,35)(29,36)(30,38)(31,39)(32,40)(33,37);(9,13)(10,11)(12,16)(14,21)(15,17)(18,19)(20,23)(22,24)(25,29)(26,35)(27,34)(28,36)(30,33)(31,40)(32,39)(37,38);(9,14,15,22)(10,18,16,23)(11,19,12,20)(13,21,17,24)(25,30,31,27)(26,36,37,32)(28,35,39,38)(29,34,40,33)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8603921);