#7746736

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

5378 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=h,e^2=g,f^2=i,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, 7653932);