#7971670

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

4276 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)(11,17)(12,16)(13,15)(14,19)(18,23)(20,22)(21,24)(25,26)(27,36)(28,30)(29,40)(31,33)(32,37)(34,39)(35,38);(1,3,5,8)(2,6,4,7)(9,11)(10,14)(12,18)(13,19)(15,17)(16,21)(20,24)(22,23)(25,27,31,37)(26,32,33,36)(28,35,39,40)(29,34,38,30);(10,15)(14,17)(16,22)(21,23)(26,33)(27,37)(28,39)(29,38);(1,2,5,4)(3,6,8,7)(9,10)(11,14)(12,16)(13,15)(17,19)(18,21)(20,22)(23,24)(25,28)(26,34)(27,35)(29,36)(30,33)(31,39)(32,38)(37,40);(1,4,5,2)(3,7,8,6)(9,11)(10,14)(12,18)(13,19)(15,17)(16,21)(20,24)(22,23)(25,29)(26,35)(27,34)(28,36)(30,37)(31,38)(32,39)(33,40);(9,12,13,20)(10,16,15,22)(11,18,19,24)(14,21,17,23)(25,30,31,34)(26,28,33,39)(27,38,37,29)(32,40,36,35)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7878866);