#8346324

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

4657 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=hi,d^2=gi,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,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, 8253520);