#8071377

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

4907 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=g,e^2=g,f^2=h,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, 7978573);