#483108

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	C₂³×D₈	C₄
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

1116 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:

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

Polycyclic presentation:

2^9|a^2=f,e^2=hi,f^2=gh,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	#72308704, #72308709, #72308716, …
maximal normal subgroup of	#72308704, #72308709, #72308716, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390304);