#409673

order	512
nilpotency class	3
isoclinism	isoclinic class of #396117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2571
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

590 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 316869);