#8345937

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

4626 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=h,d^2=hi,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8253133);