#8460715

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

4947 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)(3,7)(4,6)(5,8)(9,10,13,16)(11,12,15,14)(18,24)(19,27)(20,29)(26,31);(1,3)(2,5)(4,8)(6,7)(9,11,13,15)(10,12,16,14)(17,18,23,24)(19,22,27,32)(20,30,29,25)(21,26,28,31);(9,11)(10,14)(12,16)(13,15)(17,19)(18,22)(20,28)(21,29)(23,27)(24,32)(25,26)(30,31);(1,2,4,6)(3,5,8,7)(9,11)(10,12)(13,15)(14,16)(17,20,23,29)(18,25,24,30)(19,28,27,21)(22,26,32,31);(1,3)(2,5)(4,8)(6,7)(17,21,23,28)(18,26,24,31)(19,29,27,20)(22,25,32,30);(1,2)(3,5)(4,6)(7,8)(9,12)(10,15)(11,16)(13,14)(17,22,23,32)(18,27,24,19)(20,31,29,26)(21,25,28,30)

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=g,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8367911);