#7870474

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

4491 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)(9,10,14,17)(11,15,19,13)(12,20,22,16)(18,21,24,23)(26,32)(27,29)(34,39)(35,37);(1,2)(3,5)(4,6)(7,8)(25,26)(27,28)(29,33)(30,34)(31,32)(35,36)(37,40)(38,39);(9,11,14,19)(10,13,17,15)(12,18,22,24)(16,21,20,23)(25,27,31,29)(26,33,32,28)(30,35,38,37)(34,40,39,36);(9,11)(10,15)(12,18)(13,17)(14,19)(16,23)(20,21)(22,24)(25,28,31,33)(26,27,32,29)(30,36,38,40)(34,35,39,37);(9,12,14,22)(10,16,17,20)(11,18,19,24)(13,21,15,23)(25,29,31,27)(26,33,32,28)(30,35,38,37)(34,36,39,40);(1,3,4,7)(2,5,6,8)(9,13,14,15)(10,11,17,19)(12,21,22,23)(16,18,20,24)(25,30)(26,34)(27,35)(28,36)(29,37)(31,38)(32,39)(33,40)

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=i,e^2=h,f^2=gh,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, 7777670);