#7971686

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	#2540	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4243 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=g,d^2=h,e^2=gh,f^2=gh,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, 7878882);