#8526242

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

4364 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=hi,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8433438);