#8664337

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

4649 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gh,d^2=gh,e^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, 8571533);