#8968161

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

5444 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=ghi,d^2=gi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8875357);