#6439617

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	#23438	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2242 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:

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

Polycyclic presentation:

2^9|a^2=fi,d^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,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, 6346813);