#8664404

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

4315 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8571600);