#8174321

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

4613 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=gi,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8081517);