#7900781

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

4241 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=i,e^2=i,f^2=hi,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, 7807977);