#8361907

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

4949 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)(11,15)(12,14)(13,16)(17,18,23,27)(19,22,29,24)(20,26,21,25)(28,32,31,30);(1,2)(3,7)(4,5)(6,8)(9,10,12,14)(11,13,16,15)(17,19)(18,24)(20,28)(21,31)(22,27)(23,29)(25,32)(26,30);(2,5)(6,7)(9,10)(11,13)(12,14)(15,16)(17,20,23,21)(18,25,27,26)(19,28,29,31)(22,30,24,32);(3,8)(6,7)(9,11,12,16)(10,13,14,15)(17,21,23,20)(18,26,27,25)(19,28,29,31)(22,30,24,32);(1,3,4,8)(2,6,5,7)(17,22)(18,19)(20,30)(21,32)(23,24)(25,28)(26,31)(27,29);(1,2)(3,6)(4,5)(7,8)(17,19)(18,22)(20,31)(21,28)(23,29)(24,27)(25,32)(26,30)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=g,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=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, 8269103);