#8549673

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

5172 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)(3,7)(9,10,12,14)(11,13,15,16)(17,18,22,23)(19,20,26,24)(21,25,30,31)(27,28,29,32);(1,2,4,5)(3,6,7,8)(11,15)(13,16)(18,23)(19,26)(25,31)(27,29);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(18,23)(20,24)(25,31)(28,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,23)(21,29)(22,26)(25,28)(27,30)(31,32);(1,3,4,7)(2,6,5,8)(9,10)(11,13)(12,14)(15,16)(17,20,22,24)(18,19,23,26)(21,28,30,32)(25,27,31,29);(17,21,22,30)(18,25,23,31)(19,27,26,29)(20,28,24,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=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, 8456869);