#7985156

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

4726 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)(3,4)(5,7)(6,8)(9,10,13,12)(11,15,16,14)(17,18,23,20)(19,27,29,24)(21,30,32,25)(22,26,28,31);(1,3)(2,4)(5,8)(6,7)(9,11,13,16)(10,14,12,15)(17,19,23,29)(18,24,20,27)(21,28,32,22)(25,31,30,26);(1,4,6,5)(2,3,8,7)(9,10,13,12)(11,14,16,15)(18,20)(24,27)(25,30)(26,31);(1,3)(2,5)(4,8)(6,7)(17,20,23,18)(19,27,29,24)(21,30,32,25)(22,31,28,26);(1,3)(2,4)(5,8)(6,7)(17,21)(18,25)(19,28)(20,30)(22,29)(23,32)(24,31)(26,27);(1,5,6,4)(2,7,8,3)(9,12,13,10)(11,15,16,14)(17,22,23,28)(18,26,20,31)(19,21,29,32)(24,25,27,30)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,d^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,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, 7892352);