#8037061

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

4589 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)(9,10,14,15)(11,17,18,12)(13,21,23,16)(19,20,24,22)(25,26)(27,30)(28,35)(29,32)(31,36)(33,39)(34,38)(37,40);(1,2)(3,6)(4,7)(5,8)(9,11,14,18)(10,12,15,17)(13,19,23,24)(16,20,21,22)(25,27,29,34)(26,30,32,38)(28,33,36,40)(31,37,35,39);(1,3,5,4)(2,6,8,7)(10,15)(11,18)(16,21)(19,24)(25,26,29,32)(27,30,34,38)(28,31,36,35)(33,37,40,39);(9,10)(11,12)(13,16)(14,15)(17,18)(19,20)(21,23)(22,24)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(9,12)(10,11)(13,22)(14,17)(15,18)(16,24)(19,21)(20,23)(25,27)(26,30)(28,33)(29,34)(31,37)(32,38)(35,39)(36,40);(1,4,5,3)(2,7,8,6)(9,13,14,23)(10,16,15,21)(11,19,18,24)(12,20,17,22)(25,28)(26,31)(27,33)(29,36)(30,37)(32,35)(34,40)(38,39)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7944257);