#7852748

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

4792 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)(3,8)(4,6)(5,7)(9,10,13,12)(11,15,16,14)(19,27)(21,29)(24,28)(25,30);(1,3,5,4)(2,6,7,8)(9,11,13,16)(10,14,12,15)(17,18,23,20)(19,28,27,24)(21,25,29,30)(22,26,32,31);(1,2)(3,8)(4,6)(5,7)(17,19,23,27)(18,24,20,28)(21,22,29,32)(25,31,30,26);(17,20,23,18)(19,24,27,28)(21,30,29,25)(22,31,32,26);(1,4,5,3)(2,6,7,8)(9,12,13,10)(11,15,16,14)(17,21)(18,25)(19,22)(20,30)(23,29)(24,31)(26,28)(27,32);(1,2)(3,6)(4,8)(5,7)(17,22)(18,26)(19,29)(20,31)(21,27)(23,32)(24,25)(28,30)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=h,e^2=g,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, 7759944);