#338313

order	512
nilpotency class	3
isoclinism	isoclinic class of #332071
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×D₈	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

620 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)(11,14)(12,15)(13,17)(16,20)(18,21)(19,23)(22,24)(25,26,29,30)(27,33,35,31)(28,32,37,38)(34,36,40,39);(1,3,5,2)(4,7,8,6)(9,11,13,19)(10,14,17,23)(12,18,20,24)(15,21,16,22)(25,27,29,35)(26,31,30,33)(28,36,37,39)(32,34,38,40);(1,4,5,8)(2,6,3,7)(10,15)(12,20)(14,21)(16,17)(18,24)(22,23)(25,28,29,37)(26,32,30,38)(27,34,35,40)(31,39,33,36);(2,3)(6,7)(10,15)(12,20)(14,21)(16,17)(18,24)(22,23)(25,26,29,30)(27,33,35,31)(28,32,37,38)(34,36,40,39)

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=e,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defi,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

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 245509);