#8289460

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

5031 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=hi,e^2=h,f^2=g,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, 8196656);