#7987317

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

4342 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)(3,5,7,8)(9,10)(11,12)(13,17)(14,15)(16,20)(18,19)(21,22)(23,24)(26,30)(27,33)(32,36)(35,37);(1,2)(3,5)(4,6)(7,8)(25,26)(27,31)(28,36)(29,30)(32,38)(33,34)(35,40)(37,39);(10,15)(11,16)(17,22)(18,23)(25,26,29,30)(27,34,33,31)(28,32,38,36)(35,40,37,39);(9,11,14,16)(10,12,15,20)(13,18,21,23)(17,19,22,24)(25,26)(27,31)(28,32)(29,30)(33,34)(35,39)(36,38)(37,40);(1,3,4,7)(2,5,6,8)(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,22)(21,24)(25,27,29,33)(26,31,30,34)(28,37,38,35)(32,40,36,39);(9,13,14,21)(10,17,15,22)(11,18,16,23)(12,19,20,24)(25,28)(26,32)(27,35)(29,38)(30,36)(31,39)(33,37)(34,40)

Polycyclic presentation:

2^9|a^2=gi,c^2=i,d^2=hi,e^2=g,f^2=hi,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, 7894513);