#8271278

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

4482 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=hi,d^2=ghi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8178474);