#474244

order	512
nilpotency class	3
isoclinism	isoclinic class of #455641
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

543 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 381440);