#8689873

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

4564 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)(11,12)(13,17)(14,15)(16,18)(19,20)(21,22)(23,24)(26,28)(27,33)(32,36)(35,37);(1,2)(3,7)(4,6)(5,8)(10,15)(11,18)(17,24)(19,21)(25,26,30,28)(27,31,33,34)(29,32,38,36)(35,39,37,40);(1,3)(2,5)(4,8)(6,7)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,40)(37,38);(2,6)(5,7)(9,11)(10,16)(12,15)(13,21)(14,18)(17,20)(19,23)(22,24)(25,28,30,26)(27,34,33,31)(29,32,38,36)(35,39,37,40);(1,3)(2,7)(4,8)(5,6)(9,12,14,16)(10,11,15,18)(13,22,23,20)(17,21,24,19)(25,27)(26,31)(28,34)(29,37)(30,33)(32,40)(35,38)(36,39);(1,2)(3,5)(4,6)(7,8)(9,13,14,23)(10,17,15,24)(11,19,18,21)(12,20,16,22)(25,29,30,38)(26,32,28,36)(27,35,33,37)(31,39,34,40)

Polycyclic presentation:

2^9|b^2=h,d^2=h,e^2=gh,f^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, 8597069);