#8402405

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

4844 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=h,d^2=i,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8309601);