#8424809

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

4450 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=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, 8332005);