#8688198

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

4199 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)(9,10)(11,22)(12,18)(13,17)(14,19)(15,20)(16,21)(23,24)(26,30)(27,33)(28,36)(38,40);(2,5)(6,7)(9,11,16,13)(10,17,21,22)(12,20,23,14)(15,18,19,24)(25,26,29,30)(27,34,33,31)(28,32,36,37)(35,40,39,38);(1,2)(3,6)(4,5)(7,8)(9,12,16,23)(10,18,21,24)(11,20,13,14)(15,17,19,22)(25,26)(27,31)(28,32)(29,30)(33,34)(35,38)(36,37)(39,40);(1,3,4,8)(2,6,5,7)(9,13,16,11)(10,17,21,22)(12,14,23,20)(15,18,19,24)(25,27,29,33)(26,31,30,34)(28,35,36,39)(32,38,37,40);(9,14)(10,19)(11,12)(13,23)(15,21)(16,20)(17,24)(18,22)(25,28,29,36)(26,32,30,37)(27,35,33,39)(31,38,34,40);(9,15,16,19)(10,20,21,14)(11,18,13,24)(12,17,23,22)

Polycyclic presentation:

2^9|b^2=ghi,c^2=gi,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,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, 8595394);