#8136789

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

5006 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8043985);