#6431900

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	C₂×#3449	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4814 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)(6,7)(9,10)(11,16)(12,25)(13,19)(14,20)(15,18)(17,28)(21,37)(22,32)(23,33)(24,31)(26,40)(27,35)(29,36)(30,38)(34,39);(5,6)(7,8)(9,11)(10,16)(12,26)(13,22)(14,23)(15,24)(17,34)(18,31)(19,32)(20,33)(21,27)(25,40)(28,39)(29,38)(30,36)(35,37);(2,4)(5,7,8,6)(9,12)(10,17)(11,21)(13,27)(14,28)(15,29)(16,30)(18,35)(19,25)(20,36)(22,38)(23,26)(24,39)(31,40)(32,34)(33,37);(10,18)(16,31)(17,35)(19,20)(25,36)(30,40)(32,33)(34,37);(9,13,15,14)(10,19,18,20)(11,22,24,23)(12,27,29,28)(16,32,31,33)(17,25,35,36)(21,38,39,26)(30,34,40,37)

Polycyclic presentation:

2^9|c^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,h^a=hi,c^b=ch,g^b=gi,h^b=hi,g^c=gi,h^c=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, 6339096);