#497809

order	512
nilpotency class	3
isoclinism	isoclinic class of #188770
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1770 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

437 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405005);