#8677310

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	#40054	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4591 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8584506);