#9027851

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

5062 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)(3,7)(4,5)(6,8)(17,18,24,22)(19,27,31,25)(20,21,26,23)(28,29,32,30);(3,8)(6,7)(9,10,13,12)(11,15,16,14)(17,19,24,31)(18,25,22,27)(20,32,26,28)(21,29,23,30);(1,3,5,8)(2,6,4,7)(9,11,13,16)(10,14,12,15)(17,20)(18,23)(19,28)(21,22)(24,26)(25,29)(27,30)(31,32);(3,8)(6,7)(11,16)(14,15)(17,21,24,23)(18,20,22,26)(19,29,31,30)(25,32,27,28);(1,3)(2,7)(4,6)(5,8)(9,11)(10,15)(12,14)(13,16)(17,22,24,18)(19,25,31,27)(20,21,26,23)(28,29,32,30);(1,4,5,2)(3,6,8,7)(9,12,13,10)(11,14,16,15)(17,23,24,21)(18,26,22,20)(19,30,31,29)(25,28,27,32)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gi,d^2=g,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8935047);