#8658120

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	#2430	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4309 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=gh,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, 8565316);