#7885644

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

4738 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)(3,6)(4,7)(5,8)(9,10,13,12)(11,15,16,14);(1,3)(2,6)(4,8)(5,7)(9,11,13,16)(10,14,12,15)(17,18)(19,25)(20,24)(21,26)(22,27)(23,30)(28,29)(31,32);(1,2,5,8)(3,4,7,6)(17,19)(18,23)(20,28)(21,29)(22,30)(24,31)(25,27)(26,32);(1,3)(2,6)(4,8)(5,7)(9,10,13,12)(11,14,16,15)(17,20,22,26)(18,24,27,21)(19,28,30,32)(23,31,25,29);(1,3)(2,4)(5,7)(6,8)(17,19)(18,25)(20,28)(21,31)(22,30)(23,27)(24,29)(26,32);(1,4)(2,7)(3,8)(5,6)(9,12,13,10)(11,15,16,14)(17,21,22,24)(18,26,27,20)(19,29,30,31)(23,32,25,28)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7792840);