#8665394

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4544 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=gh,f^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, 8572590);