#7868920

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4529 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)(11,12)(13,17)(14,16)(15,19)(18,20)(21,23)(22,24)(25,26)(27,33)(28,34)(29,32)(30,31)(35,37)(36,40)(38,39);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,16)(13,18)(14,19)(17,22)(20,23)(21,24)(25,27)(26,28)(29,35)(30,34)(31,33)(32,36)(37,39)(38,40);(10,16)(12,15)(17,23)(20,22)(26,31)(27,34)(32,39)(35,40);(9,12,14,15)(10,11,16,19)(13,20,21,22)(17,18,23,24)(25,26)(27,28)(29,32)(30,31)(33,34)(35,36)(37,40)(38,39);(1,2,5,4)(3,6,8,7)(9,11)(10,15)(12,16)(13,18)(14,19)(17,22)(20,23)(21,24)(25,28)(26,27)(29,37)(30,33)(31,34)(32,40)(35,39)(36,38);(1,4,5,2)(3,7,8,6)(9,13,14,21)(10,17,16,23)(11,18,19,24)(12,20,15,22)(25,29,30,38)(26,32,31,39)(27,35,34,40)(28,36,33,37)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,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, 7776116);