#8968226

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

4678 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=i,d^2=hi,e^2=g,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, 8875422);