#8599360

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	#37824	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4385 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)(11,13,14,16)(12,20,23,18)(19,24,21,22)(25,26)(27,32)(28,29)(30,35)(31,33)(34,39)(36,40)(37,38);(2,5)(6,8)(9,11,15,14)(10,16,17,13)(12,21,23,19)(18,22,20,24)(25,27,31,36)(26,32,33,40)(28,37,34,35)(29,38,39,30);(1,2,4,5)(3,6,7,8)(10,17)(13,16)(18,20)(22,24)(26,33)(27,36)(29,39)(35,37);(3,7)(6,8)(9,12,15,23)(10,18,17,20)(11,19,14,21)(13,22,16,24)(25,28,31,34)(26,29,33,39)(27,35,36,37)(30,40,38,32);(1,2)(3,6)(4,5)(7,8)(9,13,15,16)(10,11,17,14)(12,22,23,24)(18,19,20,21)(25,29)(26,34)(27,30)(28,33)(31,39)(32,37)(35,40)(36,38);(1,3)(2,6)(4,7)(5,8)(9,14,15,11)(10,16,17,13)(12,19,23,21)(18,22,20,24)(25,30)(26,35)(27,29)(28,32)(31,38)(33,37)(34,40)(36,39)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ghi,d^2=h,e^2=g,f^2=g,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, 8506556);