#7947543

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×#553	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4206 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=ghi,d^2=g,e^2=ghi,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7854739);