#6439762

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	#23888	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2074 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)(9,10,15,20)(11,16,24,32)(12,18,26,35)(13,17,14,19)(21,30,23,31)(22,29,37,39)(25,34,38,40)(27,33,28,36);(2,5)(6,8)(9,11)(10,16)(12,25)(13,23)(14,21)(15,24)(17,31)(18,34)(19,30)(20,32)(22,27)(26,38)(28,37)(29,33)(35,40)(36,39);(10,17)(11,21)(13,14)(18,33)(19,20)(22,25)(23,24)(27,28)(30,31)(34,40)(35,36)(37,38);(9,12)(10,18)(11,22)(13,27)(14,28)(15,26)(16,29)(17,33)(19,36)(20,35)(21,25)(23,38)(24,37)(30,34)(31,40)(32,39);(1,3,4,7)(2,6,5,8)(12,26)(18,35)(22,37)(25,38)(27,28)(29,39)(33,36)(34,40)

Polycyclic presentation:

2^9|a^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,g^c=gi,h^c=hi,e^d=ei,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, 6346958);