#8429320

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
prime radical	#36187	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4859 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8336516);