#8716825

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

4393 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,d^2=ghi,e^2=g,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, 8624021);