#8429309

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	#36592	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5259 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=hi,d^2=g,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, 8336505);