#8701403

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	C₂×#550	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4359 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,d^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8608599);