#8662915

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

4609 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,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, 8570111);