#8174462

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

5378 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)(3,8,5,7)(10,13)(11,15)(17,18)(19,21)(20,25)(22,30)(23,24)(26,31)(27,32)(28,29);(1,3,6,5)(2,7,4,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,21)(20,23)(22,27)(24,25)(26,29)(28,31)(30,32);(1,2,6,4)(3,7,5,8)(9,10)(11,14)(12,13)(15,16)(18,24)(19,20)(26,30)(27,28);(2,4)(7,8)(9,10,12,13)(11,14,15,16)(17,20,23,19)(18,25,24,21)(22,28,31,27)(26,32,30,29);(1,4,6,2)(3,8,5,7)(9,10)(11,14)(12,13)(15,16)(17,21,23,25)(18,20,24,19)(22,29,31,32)(26,28,30,27);(1,5,6,3)(2,8,4,7)(9,11,12,15)(10,14,13,16)(17,22,23,31)(18,26,24,30)(19,27,20,28)(21,29,25,32)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=ghi,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8081658);