#8688205

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

4652 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)(3,6)(4,7)(5,8)(10,13)(11,12)(18,22)(20,27)(23,28)(26,31);(1,3)(2,4)(5,7)(6,8)(9,10,14,13)(11,16,12,15)(19,25)(23,28)(26,31)(30,32);(9,11,14,12)(10,15,13,16)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32);(1,3)(2,6)(4,8)(5,7)(9,12,14,11)(10,16,13,15)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32);(1,3)(2,6)(4,8)(5,7)(9,13,14,10)(11,16,12,15)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32);(1,4,5,6)(2,7,8,3)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi

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, 8595401);