#7973803

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	#59689	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4480 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=gh,e^2=h,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, 7880999);