#498075

order	512
nilpotency class	3
isoclinism	isoclinic class of #189517
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

422 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)(9,10,14,22,15,21,31,37,16,23,30,18,13,20,12,19)(11,24,27,39,28,17,25,33,29,35,32,34,26,36,40,38);(1,2,4,5)(3,7,8,6)(9,11,16,29)(10,17)(12,25,31,40)(13,26,15,28)(14,32,30,27)(18,33)(19,34)(20,24)(21,35)(22,38)(23,36)(37,39);(9,12,13,30,16,31,15,14)(10,18,21,19,23,22,20,37)(11,25,26,27,29,40,28,32)(17,33,35,34,36,38,24,39);(1,3,4,8)(2,6,5,7)(9,12,13,30,16,31,15,14)(10,19,20,18,23,37,21,22)(11,25,26,27,29,40,28,32)(17,34,24,33,36,39,35,38)

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405271);