#8073126

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

4736 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7980322);