#8665357

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
prime radical	#51522	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4783 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)(17,18)(19,29)(20,26)(21,23)(22,27)(24,28)(25,32)(30,31);(1,3)(2,6)(4,7)(5,8)(9,10)(11,16)(12,14)(13,15)(17,19,24,32)(18,25,28,29)(20,30,21,22)(23,31,26,27);(1,2)(3,4)(5,7)(6,8)(9,11,13,14)(10,12,15,16)(17,20,24,21)(18,26,28,23)(19,30,32,22)(25,27,29,31);(3,8)(4,6)(9,10)(11,12)(13,15)(14,16)(17,21,24,20)(18,26,28,23)(19,22,32,30)(25,27,29,31);(1,2)(3,6)(4,8)(5,7)(9,10)(11,12)(13,15)(14,16)(17,22)(18,27)(19,20)(21,32)(23,25)(24,30)(26,29)(28,31);(1,4,5,6)(2,3,7,8)(9,12)(10,14)(11,15)(13,16)(17,23,24,26)(18,21,28,20)(19,31,32,27)(22,29,30,25)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=ghi,d^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, 8572553);