#8689223

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	#40692	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4386 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8596419);