#6439771

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	#23887	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)(11,16)(12,18)(13,17)(14,19)(15,20)(21,31)(22,29)(23,30)(24,32)(25,34)(26,33)(27,36)(28,35)(37,39)(38,40);(2,5)(6,8)(9,11)(10,16)(12,25)(13,23)(14,21)(15,24)(17,30)(18,34)(19,31)(20,32)(22,26)(27,37)(28,38)(29,33)(35,40)(36,39);(10,17)(11,21)(13,14)(18,33)(19,20)(22,25)(23,24)(26,28)(30,31)(34,39)(35,36)(37,38);(9,12,15,27)(10,18,20,36)(11,22,24,38)(13,26,14,28)(16,29,32,40)(17,33,19,35)(21,25,23,37)(30,39,31,34);(1,3,4,7)(2,6,5,8)(9,13,15,14)(10,17,20,19)(11,23,24,21)(12,26,27,28)(16,30,32,31)(18,33,36,35)(22,37,38,25)(29,39,40,34)

Polycyclic presentation:

2^9|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, 6346967);