#8720360

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

4841 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=hi,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8627556);