#8107567

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	#2433	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4434 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014763);