#483322

order	512
nilpotency class	4
isoclinism	isoclinic class of #365
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#106
commutator	C₈	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×#106
prime radical	C₂⁴	#106
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

540 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,d^2=i,e^2=g,f^2=h,g^2=i,b^a=be,e^a=eg,g^a=gi,c^b=ci,f^b=fi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216671, #72216710, #72216720, …
maximal normal subgroup of	#72216671, #72216710, #72216720, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390518);