#8814045

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

4582 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8721241);