#8946232

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

4680 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=ghi,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=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, 8853428);