#8696749

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

4526 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=i,d^2=gh,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, 8603945);