#8891274

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

4533 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8798470);