#6453839

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	#62793	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3115 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)(13,26)(14,18)(15,21)(17,28)(20,36)(22,34)(23,39)(24,35)(25,33)(27,30)(29,32)(31,38)(37,40);(9,11,14,22)(10,15,18,30)(12,23,25,20)(13,27,28,21)(16,31,33,40)(17,34,26,19)(24,37,29,38)(32,36,35,39);(2,5)(6,8)(13,28)(15,30)(17,26)(21,27)(24,29)(31,40)(32,35)(37,38);(9,12,14,25)(10,16,18,33)(11,20,22,23)(13,29,28,24)(15,31,30,40)(17,35,26,32)(19,36,34,39)(21,38,27,37);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|b^2=i,d^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	#72317850
maximal normal subgroup of	#72317850

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6361035);