#8284170

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

5154 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)(4,6)(5,7)(9,10,12,14)(11,13,15,16)(17,18)(19,20)(21,29)(22,23)(24,31)(25,28)(26,30)(27,32);(1,2,6,4)(3,8,5,7)(11,15)(13,16)(17,19,22,25)(18,20,23,28)(21,26,31,32)(24,27,29,30);(1,2,6,4)(3,7,5,8)(9,10)(11,13)(12,14)(15,16)(18,23)(19,25)(24,29)(26,32);(1,3,6,5)(2,7,4,8)(9,11)(10,13)(12,15)(14,16)(17,20)(18,19)(21,30)(22,28)(23,25)(24,32)(26,29)(27,31);(1,4,6,2)(3,7,5,8)(9,10)(11,13)(12,14)(15,16)(17,21)(18,24)(19,26)(20,27)(22,31)(23,29)(25,32)(28,30);(1,5,6,3)(2,7,4,8)(17,18)(19,20)(21,24)(22,23)(25,28)(26,27)(29,31)(30,32)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=g,d^2=g,e^2=g,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, 8191366);