#8167409

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	#40493	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4362 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=gh,d^2=gi,e^2=gi,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, 8074605);