#496465

order	512
nilpotency class	3
isoclinism	isoclinic class of #3192
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×C₂²⋊C₈
prime radical	#3444	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

810 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=e,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cei,f^c=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403661);