#338859

order	512
nilpotency class	3
isoclinism	isoclinic class of #331655
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

517 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)(9,10,12,17)(11,18,19,14)(13,15,21,16)(20,23,24,22)(25,26,29,32)(27,30,35,39)(28,31,36,40)(33,37,34,38);(1,3,5,7)(2,6,8,4)(9,11)(10,14)(12,19)(13,20)(15,22)(16,23)(17,18)(21,24)(25,27,29,35)(26,30,32,39)(28,33,36,34)(31,37,40,38);(1,4,5,6)(2,7,8,3)(10,15)(13,21)(14,22)(16,17)(18,23)(20,24)(27,33)(28,36)(30,37)(31,40)(34,35)(38,39);(2,8)(3,7)(10,16)(11,19)(13,21)(14,22)(15,17)(18,23)

Polycyclic presentation:

2^9|a^2=ei,b^2=ehi,c^2=eh,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, 246055);