#8429313

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	#2537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4653 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:

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

Polycyclic presentation:

2^9|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, 8336509);