#8273428

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

4448 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,e^2=h,f^2=g,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, 8180624);