#8720827

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4493 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8628023);