#8429655

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

4957 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8336851);