#8936944

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	#47992	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4431 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)(11,19)(12,15)(13,17)(14,18)(16,22)(20,24)(21,23)(25,26,31,29)(27,28,35,32)(30,37,39,33)(34,40,38,36);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,19)(13,20)(14,21)(15,22)(17,18)(23,24)(25,27,31,35)(26,32,29,28)(30,34,39,38)(33,40,37,36);(2,5)(3,7)(13,21)(14,20)(17,23)(18,24)(25,27,31,35)(26,32,29,28)(30,38,39,34)(33,36,37,40);(3,7)(6,8)(9,12,15,10)(11,19,22,16)(13,17,21,23)(14,24,20,18)(25,28)(26,27)(29,35)(30,36)(31,32)(33,34)(37,38)(39,40);(1,3)(2,6)(4,7)(5,8)(9,13,15,21)(10,17,12,23)(11,20,22,14)(16,18,19,24)(25,29,31,26)(27,32,35,28)(30,37,39,33)(34,40,38,36);(1,2)(3,6)(4,5)(7,8)(9,14,15,20)(10,18,12,24)(11,21,22,13)(16,17,19,23)(25,30)(26,33)(27,34)(28,36)(29,37)(31,39)(32,40)(35,38)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8844140);