#8592242

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	#40689	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4389 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=g,d^2=g,e^2=gh,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, 8499438);