#315197

order	512
nilpotency class	3
isoclinism	isoclinic class of #288899
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

557 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)(3,5)(4,6)(7,8)(9,10,14,16)(11,19,21,17)(12,15,22,24)(13,23,20,18)(25,26,28,33,29,32,37,31)(27,30,35,39,36,38,34,40);(1,3,4,7)(2,5,6,8)(9,11,20,22,14,21,13,12)(10,15,23,17,16,24,18,19)(25,27,28,35,29,36,37,34)(26,30,33,39,32,38,31,40);(2,6)(5,8)(10,16)(11,21)(12,22)(18,23)(26,31)(27,34)(28,37)(30,38)(32,33)(35,36);(9,12,13,21,14,22,20,11)(10,17,18,15,16,19,23,24)(26,32)(30,38)(31,33)(39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=ef,d^2=e,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,d^c=dh,f^c=fi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 222393);