#8700456

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4296 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=hi,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8607652);