#8899321

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	#46367	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4725 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=gh,e^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8806517);