#8703437

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

5800 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=h,e^2=gh,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, 8610633);