#8944500

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

4686 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=ghi,d^2=gh,e^2=g,f^2=gh,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, 8851696);