#8260788

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

4759 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=i,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8167984);