#9062112

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

4778 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gh,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=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, 8969308);