#8390853

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

5463 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8298049);