#483123

order	512
nilpotency class	4
isoclinism	isoclinic class of #482075
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1078 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,e^2=hi,f^2=ghi,h^2=i,b^a=be,c^a=cg,d^a=di,e^a=eg,e^b=eh,f^b=fgh,h^b=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310135, #72310139, #72310140, …
maximal normal subgroup of	#72310135, #72310139, #72310140, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390319);