#8689922

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

4520 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=g,f^2=h,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, 8597118);