#6439619

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	#23439	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2291 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)(9,10,15,19)(11,16,23,32)(12,18,27,35)(13,17,14,20)(21,30,24,31)(22,29,37,39)(25,34,38,40)(26,33,28,36);(2,5)(6,8)(9,11)(10,16)(12,25)(13,24)(14,21)(15,23)(17,31)(18,34)(19,32)(20,30)(22,26)(27,38)(28,37)(29,33)(35,40)(36,39);(10,17)(11,21)(13,14)(18,33)(19,20)(22,25)(23,24)(26,28)(30,31)(34,40)(35,36)(37,38);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,22)(13,26)(14,28)(15,27)(16,29)(17,33)(19,35)(20,36)(21,25)(23,37)(24,38)(30,34)(31,40)(32,39);(10,19)(11,23)(17,20)(18,35)(21,24)(22,37)(25,38)(33,36)

Polycyclic presentation:

2^9|a^2=fi,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, 6346815);