#8659368

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	#56316	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4458 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8566564);