#8841740

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

4341 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=ghi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

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, 8748936);