#8827971

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

5294 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,d^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8735167);