#7856906

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

5363 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=g,d^2=h,e^2=gh,f^2=gh,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, 7764102);