#8161260

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	#37940	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4435 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,d^2=i,e^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^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, 8068456);