#7979826

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4357 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)(11,14)(12,19)(13,21)(15,16)(17,22)(18,23)(20,24)(26,31)(28,37)(32,33)(34,40);(1,3,5,8)(2,6,4,7)(9,11,15,12)(10,14,16,19)(13,18,22,20)(17,24,21,23)(25,26,30,31)(27,33,36,32)(28,29,37,39)(34,38,40,35);(1,2,5,4)(3,6,8,7)(10,16)(11,12)(13,22)(23,24)(25,27,30,36)(26,32,31,33)(28,34,37,40)(29,35,39,38);(1,4,5,2)(3,7,8,6)(9,12,15,11)(10,14,16,19)(13,20,22,18)(17,24,21,23)(25,28)(26,29)(27,34)(30,37)(31,39)(32,35)(33,38)(36,40);(9,13)(10,17)(11,18)(12,20)(14,24)(15,22)(16,21)(19,23)(25,27)(26,33)(28,34)(29,38)(30,36)(31,32)(35,39)(37,40);(9,14)(10,11)(12,16)(13,23)(15,19)(17,20)(18,21)(22,24)(25,29)(26,28)(27,35)(30,39)(31,37)(32,34)(33,40)(36,38)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,d^2=gi,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, 7887022);