#8664436

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

4563 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=gh,e^2=h,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, 8571632);