#7985554

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

4836 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=i,d^2=g,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, 7892750);