#8186290

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	#38949	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4625 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)(11,13)(12,15)(14,22)(16,18)(17,24)(19,20)(21,23)(25,26)(27,37)(28,40)(29,35)(30,34)(31,36)(32,39)(33,38);(1,2)(3,7)(4,5)(6,8)(9,11,16,15)(10,13,18,12)(14,19,24,23)(17,21,22,20)(25,27)(26,32)(28,29)(30,38)(31,39)(33,35)(34,40)(36,37);(2,5)(6,7)(9,12,16,13)(10,11,18,15)(14,20,24,21)(17,19,22,23)(25,28,31,38)(26,33,36,40)(27,29,39,30)(32,35,37,34);(1,3,4,8)(2,6,5,7)(9,13,16,12)(10,11,18,15)(14,21,24,20)(17,19,22,23)(25,29,31,30)(26,34,36,35)(27,28,39,38)(32,40,37,33);(9,14,16,24)(10,17,18,22)(11,19,15,23)(12,20,13,21)(25,30,31,29)(26,35,36,34)(27,38,39,28)(32,33,37,40);(1,2)(3,6)(4,5)(7,8)(9,15,16,11)(10,12,18,13)(14,23,24,19)(17,20,22,21)(25,26)(27,32)(28,40)(29,34)(30,35)(31,36)(33,38)(37,39)

Polycyclic presentation:

2^9|b^2=gh,c^2=h,d^2=gi,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, 8093486);