#8526804

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

4167 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gi,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=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, 8434000);