#345688

order	512
nilpotency class	3
isoclinism	isoclinic class of #334823
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

535 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ei,c^2=hi,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 252884);