#8594872

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

4751 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8502068);