#10086365

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

4979 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=hi,d^2=i,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9993561);