#338934

order	512
nilpotency class	3
isoclinism	isoclinic class of #331655
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

522 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ehi,b^2=ehi,c^2=ehi,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 246130);