#7983578

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

4487 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=h,e^2=h,f^2=i,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, 7890774);