#7746080

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
prime radical	#35276	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4703 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)(4,6)(5,8)(9,10,13,12)(11,15,16,14);(2,4)(7,8)(9,11,13,16)(10,14,12,15)(17,18)(19,23)(20,27)(21,22)(24,28)(25,31)(26,29)(30,32);(1,2,6,4)(3,8,5,7)(9,10,13,12)(11,14,16,15)(18,22)(23,29)(24,27)(30,31);(1,3,6,5)(2,7,4,8)(9,10,13,12)(11,14,16,15)(17,19,21,26)(18,23,22,29)(20,25,28,32)(24,30,27,31);(1,4,6,2)(3,8,5,7)(17,20)(18,24)(19,25)(21,28)(22,27)(23,30)(26,32)(29,31);(1,5,6,3)(2,8,4,7)(9,12,13,10)(11,15,16,14)(17,18)(19,23)(20,24)(21,22)(25,30)(26,29)(27,28)(31,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7653276);