#8750392

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

4749 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)(4,6)(5,8)(9,10)(11,18)(12,16)(13,15)(14,17)(19,24)(20,23)(21,22)(25,26,31,34)(27,29,36,33)(28,37,30,32)(35,39,38,40);(1,2,4,6)(3,5,8,7)(9,11,14,13)(10,15,17,18)(12,19,20,21)(16,22,23,24)(25,27)(26,29)(28,38)(30,35)(31,36)(32,39)(33,34)(37,40);(1,2)(3,5)(4,6)(7,8)(12,20)(16,23)(19,21)(22,24)(25,28,31,30)(26,32,34,37)(27,35,36,38)(29,40,33,39);(1,3)(2,5)(4,8)(6,7)(9,12,14,20)(10,16,17,23)(11,19,13,21)(15,22,18,24)(25,29,31,33)(26,27,34,36)(28,39,30,40)(32,38,37,35);(9,10)(11,15)(12,16)(13,18)(14,17)(19,22)(20,23)(21,24)(25,27)(26,33)(28,35)(29,34)(30,38)(31,36)(32,39)(37,40);(9,13,14,11)(10,15,17,18)(12,19,20,21)(16,24,23,22)(25,30,31,28)(26,32,34,37)(27,35,36,38)(29,39,33,40)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8657588);