#8525900

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

5019 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8433096);