#8346900

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

4957 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=g,e^2=g,f^2=g,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, 8254096);