#8167370

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5536 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=ghi,d^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8074566);