#8968133

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

4996 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,16)(13,14)(17,18,21,24)(19,22,25,26)(20,23,28,30)(27,29,31,32);(1,2,4,6)(3,5,8,7)(9,10,13,14)(11,15,16,12)(17,18)(19,22)(20,23)(21,24)(25,26)(27,29)(28,30)(31,32);(1,2,4,6)(3,7,8,5)(9,11)(10,12)(13,16)(14,15)(19,25)(20,28)(22,26)(23,30);(1,2,4,6)(3,5,8,7)(9,12)(10,11)(13,15)(14,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,30)(29,31);(1,2)(3,5)(4,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,19,21,25)(18,22,24,26)(20,27,28,31)(23,29,30,32);(1,3,4,8)(2,5,6,7)(9,12)(10,11)(13,15)(14,16)(17,20)(18,23)(19,27)(21,28)(22,29)(24,30)(25,31)(26,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=hi,d^2=hi,e^2=gh,f^2=hi,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, 8875329);