#10087360

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	#39118	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4222 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9994556);