#8656214

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

5240 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,d^2=hi,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,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8563410);