#8126986

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4706 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,d^2=gh,e^2=h,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^c=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, 8034182);