#8272377

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

5064 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8179573);