#8526726

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

4171 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,d^2=h,e^2=gh,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, 8433922);