#8751691

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
prime radical	#2541	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4304 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)(11,19)(12,13)(14,16)(15,17)(18,23)(20,21)(22,24)(25,26,31,35)(27,33,29,32)(28,34,39,30)(36,37,38,40);(1,2,4,5)(3,6,7,8)(9,11,15,14)(10,16,17,19)(12,22,18,20)(13,21,23,24)(25,27,31,29)(26,32,35,33)(28,36,39,38)(30,37,34,40);(1,2)(3,6)(4,5)(7,8)(10,17)(12,18)(16,19)(20,22)(25,27,31,29)(26,33,35,32)(28,38,39,36)(30,37,34,40);(1,2)(3,6)(4,5)(7,8)(9,12,15,18)(10,13,17,23)(11,20,14,22)(16,24,19,21)(25,28,31,39)(26,34,35,30)(27,36,29,38)(32,40,33,37);(1,3,4,7)(2,6,5,8)(9,13)(10,18)(11,21)(12,17)(14,24)(15,23)(16,20)(19,22)(25,29,31,27)(26,32,35,33)(28,38,39,36)(30,37,34,40);(9,14,15,11)(10,16,17,19)(12,20,18,22)(13,21,23,24)(25,30,31,34)(26,28,35,39)(27,37,29,40)(32,36,33,38)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=gi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8658887);