#8720671

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

5051 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=i,d^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8627867);