#8425567

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	#34889	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4399 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8332763);