#7970278

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

4494 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,e^2=gh,f^2=g,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, 7877474);