#6439775

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437602
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	#23889	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2047 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)(11,17)(12,26)(13,19)(14,20)(15,21)(16,22)(18,29)(23,34)(24,40)(25,35)(27,38)(28,37)(30,36)(31,33)(32,39);(2,5)(6,8)(9,11)(10,17)(12,27)(13,29)(14,23)(15,24)(16,25)(18,19)(20,34)(21,40)(22,35)(26,38)(28,32)(30,31)(33,36)(37,39);(9,12,16,30)(10,18,22,37)(11,13,25,32)(14,29,15,28)(17,33,35,38)(19,34,39,40)(20,36,21,26)(23,31,24,27);(9,13,16,32)(10,19,22,39)(11,12,25,30)(14,27,15,31)(17,26,35,36)(18,34,37,40)(20,38,21,33)(23,28,24,29);(1,3,4,7)(2,6,5,8)(9,14,16,15)(10,20,22,21)(11,23,25,24)(12,28,30,29)(13,27,32,31)(17,34,35,40)(18,26,37,36)(19,38,39,33)

Polycyclic presentation:

2^9|c^2=i,d^2=i,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,h^b=hi,e^c=ei,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, 6346971);