#8127048

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

4487 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8034244);