#498087

order	512
nilpotency class	3
isoclinism	isoclinic class of #189674
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

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	C₂³	C₂²×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1770 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

400 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)(9,10,15,21,14,19,25,33,16,22,13,17,31,20,11,23)(12,26,27,38,28,18,34,35,29,36,30,39,40,37,24,32);(1,2)(3,7)(4,8)(5,6)(9,11,14,15,16,25,31,13)(10,17,20,33,22,21,19,23)(12,24,28,27,29,34,40,30)(18,32,26,39,37,35,36,38);(1,3,5,4)(2,7,6,8)(9,12,16,29)(10,18)(11,24,25,34)(13,30,15,27)(14,28,31,40)(17,32)(19,36)(20,26)(21,35)(22,37)(23,38)(33,39);(1,4,5,3)(2,7,6,8)(9,13,14,11,16,15,31,25)(10,17,19,23,22,21,20,33)(12,27,28,34,29,30,40,24)(18,35,36,39,37,32,26,38)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405283);