#8658053

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

4335 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8565249);