#8947106

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

4635 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=i,e^2=gh,f^2=g,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, 8854302);