#6431961

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	C₂³×#135	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)(13,18)(14,19)(15,20)(16,24)(17,29)(22,38)(23,33)(25,32)(27,34)(28,36)(30,35)(31,39)(37,40);(5,6)(7,8)(9,11)(10,16)(12,27)(13,23)(14,24)(15,25)(17,34)(18,21)(19,32)(20,33)(22,30)(26,38)(28,40)(29,39)(31,36)(35,37);(2,4)(6,8)(9,12,13,29,15,28,14,30)(10,17,19,26,20,35,18,36)(11,22,24,40,25,39,23,27)(16,31,21,37,33,38,32,34);(12,28)(17,35)(22,39)(26,36)(27,40)(29,30)(31,38)(34,37);(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,29,28,30)(16,21,33,32)(17,36,35,26)(22,27,39,40)(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,f^a=fi,g^a=gi,h^a=hi,c^b=ch,f^b=fi,g^b=gi,h^b=hi,d^c=di

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, 6339157);