#363103

order	512
nilpotency class	3
isoclinism	isoclinic class of #188774
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	C₂³×C₄	C₂⁴
socle	C₂²	#2283
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

683 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=g,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,h^b=hi,h^c=hi,h^d=hi

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, 270299);