#8340124

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

4843 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=i,d^2=ghi,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8247320);