#8188020

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	#42736	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4403 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=gh,d^2=hi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8095216);