#8273519

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

5127 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)(11,12)(13,14)(15,16)(17,18,23,25)(19,24,27,21)(20,26,30,32)(22,28,29,31);(1,2)(3,6)(4,5)(7,8)(10,14)(11,15)(17,19,23,27)(18,24,25,21)(20,29,30,22)(26,31,32,28);(2,5)(6,8)(9,11,13,15)(10,12,14,16)(18,25)(19,27)(22,29)(26,32);(3,7)(6,8)(9,12)(10,15)(11,14)(13,16)(17,20)(18,26)(19,22)(21,31)(23,30)(24,28)(25,32)(27,29);(1,3)(2,6)(4,7)(5,8)(17,21,23,24)(18,27,25,19)(20,28,30,31)(22,32,29,26);(1,2)(3,6)(4,5)(7,8)(17,22)(18,28)(19,20)(21,32)(23,29)(24,26)(25,31)(27,30)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8180715);