#8650377

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
prime radical	#37756	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4655 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)(3,8)(4,5)(6,7)(9,10,14,15)(11,13,12,16)(17,18,22,21)(19,25,26,23)(20,24,28,29)(27,31,32,30);(1,3,6,5)(2,4,7,8)(10,15)(11,12)(19,26)(20,28)(23,25)(24,29);(2,7)(4,8)(9,11,14,12)(10,16,15,13)(17,19,22,26)(18,23,21,25)(20,27,28,32)(24,30,29,31);(2,7)(3,5)(9,12,14,11)(10,13,15,16)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32);(1,4)(2,3)(5,7)(6,8)(9,13)(10,11)(12,15)(14,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,32)(29,31);(1,5,6,3)(2,4,7,8)(17,21,22,18)(19,25,26,23)(20,24,28,29)(27,30,32,31)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=i,d^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8557573);