#8504777

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

4266 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)(9,10,16,14)(11,20,15,17)(12,19,13,18)(21,23,22,24)(25,26)(27,35)(28,29)(30,32)(31,34)(33,40)(36,37)(38,39);(1,3,4,8)(2,5,6,7)(9,11,16,15)(10,17,14,20)(12,21,13,22)(18,23,19,24)(25,27,31,30)(26,32,34,35)(28,38,33,36)(29,37,40,39);(9,12,16,13)(10,18,14,19)(11,21,15,22)(17,23,20,24)(25,28,31,33)(26,29,34,40)(27,36,30,38)(32,39,35,37);(1,2,4,6)(3,5,8,7)(9,13,16,12)(10,19,14,18)(11,21,15,22)(17,23,20,24)(25,29)(26,33)(27,37)(28,34)(30,39)(31,40)(32,36)(35,38);(1,3)(2,5)(4,8)(6,7)(9,14,16,10)(11,17,15,20)(12,19,13,18)(21,23,22,24);(1,2)(3,5)(4,6)(7,8)(9,15,16,11)(10,17,14,20)(12,22,13,21)(18,23,19,24)(25,30,31,27)(26,32,34,35)(28,38,33,36)(29,39,40,37)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8411973);