#8172732

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

4790 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=gi,e^2=g,f^2=gi,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, 8079928);