#8463721

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	#50106	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4323 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2)(3,6)(4,7)(5,8)(10,14)(11,17)(16,22)(18,19)(25,26,32,30)(27,36,37,33)(28,38,34,29)(31,35,39,40);(3,8)(5,6)(9,10)(11,15)(12,16)(13,14)(17,21)(18,23)(19,24)(20,22)(25,27,32,37)(26,33,30,36)(28,35,34,40)(29,39,38,31);(1,3)(2,5)(4,8)(6,7)(25,28)(26,29)(27,35)(30,38)(31,36)(32,34)(33,39)(37,40);(1,3)(2,6)(4,8)(5,7)(9,11,13,17)(10,15,14,21)(12,19,20,18)(16,24,22,23)(25,29,32,38)(26,34,30,28)(27,31,37,39)(33,35,36,40);(9,10)(11,15)(12,16)(13,14)(17,21)(18,23)(19,24)(20,22)(25,30,32,26)(27,36,37,33)(28,38,34,29)(31,40,39,35);(1,2)(3,5)(4,7)(6,8)(9,12,13,20)(10,16,14,22)(11,18,17,19)(15,23,21,24)(25,31)(26,35)(27,29)(28,36)(30,40)(32,39)(33,34)(37,38)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,d^2=g,e^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,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, 8370917);