#6439621

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437608
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2237 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6346817);