#472678

order	512
nilpotency class	3
isoclinism	isoclinic class of #453980
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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	#2684
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

786 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 379874);