#8709037

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

4429 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8616233);