#474721

order	512
nilpotency class	3
isoclinism	isoclinic class of #459486
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2782
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

606 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)(3,7)(4,5)(6,8)(9,10,14,31,15,20,22,19)(11,21,13,16,24,37,30,18)(12,25,27,36,28,17,32,38)(23,35,40,33,26,39,29,34);(2,5)(6,7)(9,11,22,30,15,24,14,13)(10,16,31,37,20,18,19,21)(12,23,27,40,28,26,32,29)(17,34,36,39,25,33,38,35);(1,3,4,8)(2,6,5,7)(9,12)(10,17)(11,23)(13,29)(14,32)(15,28)(16,33)(18,34)(19,38)(20,25)(21,39)(22,27)(24,26)(30,40)(31,36)(35,37);(3,8)(6,7)(9,13,14,24,15,30,22,11)(10,18,31,21,20,16,19,37)(12,26,27,29,28,23,32,40)(17,35,38,33,25,39,36,34)

Polycyclic presentation:

2^9|a^2=ef,b^2=efgh,c^2=fg,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^c=de,e^c=ei,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381917);