#6453837

order	512
nilpotency class	3
isoclinism	isoclinic class of #23912
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#120
prime radical	#62781	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3125 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dhi,f^b=fi,f^c=fi,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257412, #72257430, #72317670, …
maximal normal subgroup of	#72257412, #72257430, #72317670, …
maximal quotient of	#72257967

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6361033);