#8688341

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

4363 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)(12,20)(15,23)(19,24)(21,22)(25,26,31,30)(27,34,38,32)(28,33,37,39)(29,35,36,40);(2,5)(6,7)(9,10)(11,16)(12,15)(13,17)(14,18)(19,21)(20,23)(22,24)(25,27)(26,32)(28,36)(29,37)(30,34)(31,38)(33,35)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,19)(14,17)(15,22)(16,18)(20,24)(21,23)(25,28,31,37)(26,33,30,39)(27,36,38,29)(32,35,34,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,20)(10,15,18,23)(11,19,17,24)(13,22,16,21)(25,27)(26,34)(28,36)(29,37)(30,32)(31,38)(33,40)(35,39);(1,3,4,8)(2,6,5,7)(9,11)(10,16)(12,19)(13,18)(14,17)(15,21)(20,24)(22,23)(25,29,31,36)(26,35,30,40)(27,37,38,28)(32,33,34,39);(9,13,14,16)(10,17,18,11)(12,21,20,22)(15,19,23,24)(25,30,31,26)(27,34,38,32)(28,39,37,33)(29,35,36,40)

Polycyclic presentation:

2^9|a^2=g,c^2=gh,d^2=hi,e^2=g,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, 8595537);