#8697582

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

4418 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=hi,d^2=gi,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, 8604778);