#6432086

order	512
nilpotency class	3
isoclinism	isoclinic class of #23400
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	#23409	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3349 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)(11,16)(12,24)(13,18)(14,27)(15,19)(17,25)(20,36)(21,30)(22,35)(23,31)(26,33)(28,34)(29,37)(32,39)(38,40);(2,5)(6,8)(9,11,15,23)(10,16,19,31)(12,20,26,38)(13,22,14,21)(17,32,34,29)(18,35,27,30)(24,36,33,40)(25,39,28,37);(9,12)(10,17)(11,20)(13,25)(14,28)(15,26)(16,29)(18,33)(19,34)(21,37)(22,39)(23,38)(24,27)(30,40)(31,32)(35,36);(1,3,4,7)(2,6,5,8);(9,13,15,14)(10,18,19,27)(11,21,23,22)(12,25,26,28)(16,30,31,35)(17,33,34,24)(20,37,38,39)(29,40,32,36)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257430
maximal normal subgroup of	#72257430
maximal quotient of	#72264531

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339282);