#498077

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

410 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)(6,7)(9,10,15,21,13,20,30,32,16,22,12,19,14,17,29,33)(11,23,24,38,25,36,39,37,27,31,28,40,26,18,34,35);(1,2,4,5)(3,7,8,6)(10,17,22,20)(12,15)(18,31,36,23)(19,33,21,32)(24,28)(29,30)(34,39)(35,38,37,40);(9,11,14,26,16,27,13,25)(10,18,20,23,22,36,17,31)(12,28,30,39,15,24,29,34)(19,37,33,40,21,35,32,38);(1,3,4,8)(2,6,5,7)(9,12,13,29,16,15,14,30)(10,19,20,33,22,21,17,32)(11,24,25,39,27,28,26,34)(18,35,23,38,36,37,31,40)

Polycyclic presentation:

2^9|a^2=g,b^2=ef,c^2=f,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, 405273);