#8665365

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

4826 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=ghi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8572561);