#8196395

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	#35293	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4529 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)(3,7)(4,5)(6,8)(10,14)(11,17)(16,19)(18,23)(25,26,30,33)(27,28,31,35)(29,32,38,40)(34,36,37,39);(2,5)(6,7)(10,14)(15,21)(16,19)(22,24)(27,31)(28,35)(34,37)(36,39);(1,2)(3,6)(4,5)(7,8)(9,10,13,14)(11,15,17,21)(12,19,20,16)(18,24,23,22)(25,27,30,31)(26,28,33,35)(29,37,38,34)(32,39,40,36);(1,3,4,8)(2,6,5,7)(25,28)(26,31)(27,33)(29,36)(30,35)(32,37)(34,40)(38,39);(9,11,13,17)(10,15,14,21)(12,18,20,23)(16,22,19,24);(9,12,13,20)(10,16,14,19)(11,18,17,23)(15,22,21,24)(25,29,30,38)(26,32,33,40)(27,34,31,37)(28,36,35,39)

Polycyclic presentation:

2^9|a^2=ghi,c^2=gi,d^2=g,e^2=h,f^2=gi,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, 8103591);