#8664574

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

5746 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:

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

Polycyclic presentation:

2^9|c^2=gi,d^2=h,e^2=g,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, 8571770);