#8289272

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
prime radical	#59674	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4665 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=hi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8196468);