#8936585

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

5094 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=gh,f^2=h,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, 8843781);