#478618

order	512
nilpotency class	3
isoclinism	isoclinic class of #476095
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

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	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

818 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:

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

Polycyclic presentation:

2^9|a^2=eg,c^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,f^c=fi,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, 385814);