#8700476

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

4756 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)(9,10,14,12)(11,15,13,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,30)(29,31);(2,5)(6,7)(9,11,14,13)(10,15,12,16)(17,19,23,21)(18,24,20,26)(22,27,30,29)(25,31,28,32);(1,2)(3,6)(4,5)(7,8)(9,11,14,13)(10,16,12,15);(1,2)(3,6)(4,5)(7,8)(9,12,14,10)(11,15,13,16)(17,20,23,18)(19,24,21,26)(22,25,30,28)(27,32,29,31);(1,3)(2,6)(4,8)(5,7)(9,13,14,11)(10,15,12,16)(17,21,23,19)(18,24,20,26)(22,27,30,29)(25,32,28,31);(17,22,23,30)(18,25,20,28)(19,27,21,29)(24,31,26,32)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=hi,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8607672);