#9007928

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

4608 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=h,d^2=g,e^2=ghi,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, 8915124);