#8274031

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	#52394	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4433 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=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, 8181227);