#8696738

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

4661 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)(3,6)(4,7)(5,8)(9,10,14,18)(11,16,12,15)(13,17,22,21)(19,24,20,23)(25,26)(27,32)(28,34)(29,38)(30,36)(31,37)(33,39)(35,40);(1,3)(2,4)(5,7)(6,8)(9,11,14,12)(10,15,18,16)(13,19,22,20)(17,23,21,24)(25,27,31,30)(26,32,37,36)(28,35,39,38)(29,34,40,33);(9,11,14,12)(10,15,18,16)(13,19,22,20)(17,23,21,24)(25,28)(26,33)(27,38)(29,36)(30,35)(31,39)(32,40)(34,37);(1,2,5,8)(3,6,7,4)(9,12,14,11)(10,16,18,15)(13,19,22,20)(17,23,21,24)(25,28)(26,34)(27,35)(29,36)(30,38)(31,39)(32,40)(33,37);(1,3)(2,6)(4,8)(5,7)(9,10)(11,15)(12,16)(13,21)(14,18)(17,22)(19,24)(20,23)(25,29)(26,35)(27,34)(28,36)(30,33)(31,40)(32,39)(37,38);(1,4,5,6)(2,7,8,3)(9,13)(10,17)(11,19)(12,20)(14,22)(15,23)(16,24)(18,21)(25,30,31,27)(26,36,37,32)(28,35,39,38)(29,34,40,33)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=hi,d^2=g,f^2=gh,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, 8603934);