#8270561

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

5187 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)(17,18)(19,25)(20,30)(21,26)(22,27)(23,24)(28,29)(31,32);(1,2,4,5)(3,6,7,8)(9,10,12,14)(11,13,16,15)(18,24)(21,27)(25,30)(28,31);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19,23,20)(18,25,24,30)(21,28,27,31)(22,32,26,29);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,20,23,19)(18,25,24,30)(21,31,27,28)(22,32,26,29);(1,3,4,7)(2,6,5,8)(17,21,23,27)(18,26,24,22)(19,28,20,31)(25,29,30,32);(17,22,23,26)(18,27,24,21)(19,29,20,32)(25,28,30,31)

Polycyclic presentation:

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

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, 8177757);