#8358571

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

4540 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,d^2=i,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^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8265767);