#6439780

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	#23890	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2076 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)(3,6)(4,5)(7,8)(9,10)(11,22)(12,27)(13,19)(14,20)(15,21)(16,33)(17,25)(18,31)(23,38)(24,37)(26,36)(28,40)(29,39)(30,35)(32,34);(2,5)(6,8)(9,11,15,25)(10,17,21,22)(12,28,30,32)(13,29,23,31)(14,24,16,26)(18,38,39,19)(20,36,33,37)(27,34,35,40);(9,12,15,30)(10,18,21,39)(11,23,25,13)(14,31,16,29)(17,34,22,40)(19,36,38,37)(20,35,33,27)(24,28,26,32);(9,13)(10,19)(11,12)(14,28)(15,23)(16,32)(17,35)(18,36)(20,40)(21,38)(22,27)(24,29)(25,30)(26,31)(33,34)(37,39);(1,3,4,7)(2,6,5,8)(9,14,15,16)(10,20,21,33)(11,24,25,26)(12,29,30,31)(13,28,23,32)(17,36,22,37)(18,27,39,35)(19,40,38,34)

Polycyclic presentation:

2^9|b^2=i,c^2=i,e^2=f,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, 6346976);