#335678

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

587 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)(3,4)(5,7)(6,8)(9,10,12,14)(11,18,19,24)(13,15,21,17)(16,20,23,22)(25,26)(27,32)(28,36)(29,31)(30,33)(34,40)(35,38)(37,39);(1,3)(2,6)(4,7)(5,8)(10,14)(11,19)(15,17)(20,22)(26,31)(27,34)(28,30)(32,39)(35,37)(38,40);(1,4,5,6)(2,3,7,8)(10,15)(13,21)(14,17)(16,24)(18,23)(20,22)(25,27,29,35)(26,32,31,38)(28,34,30,37)(33,39,36,40);(2,7)(3,8)(9,11,12,19)(10,16,14,23)(13,22,21,20)(15,24,17,18)(25,28,29,30)(26,33,31,36)(27,34,35,37)(32,39,38,40)

Polycyclic presentation:

2^9|a^2=h,c^2=eh,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, 242874);