#8898027

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

4475 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gh,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8805223);