#8322206

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

4899 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=g,d^2=g,e^2=h,f^2=gi,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, 8229402);