#9007915

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

4606 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=h,d^2=g,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=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, 8915111);