#8358584

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

5457 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)(3,5)(4,6)(7,8)(17,18,21,24)(19,22,26,30)(20,27,28,23)(25,31,32,29);(1,2,4,6)(3,5,7,8)(9,10,13,14)(11,15,16,12)(17,19,21,26)(18,22,24,30)(20,25,28,32)(23,29,27,31);(3,7)(5,8)(10,14)(12,15)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32);(1,2,4,6)(3,5,7,8)(9,11,13,16)(10,15,14,12)(17,19)(18,22)(20,25)(21,26)(23,29)(24,30)(27,31)(28,32);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,16)(14,15)(17,20,21,28)(18,23,24,27)(19,25,26,32)(22,29,30,31);(1,3)(2,5)(4,7)(6,8)(9,12)(10,11)(13,15)(14,16)

Polycyclic presentation:

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

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, 8265780);