#6431956

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂²×#575	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4786 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)(3,4)(9,10)(11,21)(12,26)(13,19)(14,20)(15,18)(16,24)(17,28)(22,38)(23,32)(25,33)(27,34)(29,35)(30,36)(31,40)(37,39);(5,6)(7,8)(9,11)(10,16)(12,27)(13,23)(14,24)(15,25)(17,34)(18,32)(19,21)(20,33)(22,29)(26,38)(28,40)(30,39)(31,36)(35,37);(2,4)(6,8)(9,12,13,28,15,30,14,29)(10,17,20,26,18,35,19,36)(11,22,24,39,25,40,23,27)(16,31,21,37,32,38,33,34);(10,18)(16,32)(17,35)(19,20)(21,33)(26,36)(31,38)(34,37);(9,13,15,14)(10,19,18,20)(11,23,25,24)(12,28,30,29)(16,21,32,33)(17,36,35,26)(22,27,40,39)(31,37,38,34)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339152);