#8505968

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

5142 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,d^2=g,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8413164);