#8172650

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

5568 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=ghi,d^2=h,e^2=h,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, 8079846);