#8678568

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	C₂×#127	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4481 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)(19,27)(21,31)(24,28)(26,32);(1,3,6,4)(2,7,5,8)(9,10)(11,16)(12,14)(13,15)(17,18,23,22)(19,28,27,24)(20,30,29,25)(21,26,31,32);(9,11,13,12)(10,14,15,16)(17,19,23,27)(18,24,22,28)(20,21,29,31)(25,26,30,32);(2,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,20,23,29)(18,25,22,30)(19,21,27,31)(24,26,28,32);(1,4,6,3)(2,7,5,8)(9,10)(11,14)(12,16)(13,15)(17,21)(18,26)(19,29)(20,27)(22,32)(23,31)(24,30)(25,28);(1,5,6,2)(3,8,4,7)(9,12,13,11)(10,14,15,16)(17,22,23,18)(19,28,27,24)(20,25,29,30)(21,26,31,32)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=i,d^2=h,e^2=g,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, 8585764);