#6444125

order	512
nilpotency class	3
isoclinism	isoclinic class of #6439025
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	C₂×#3060	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2032 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)(3,7)(4,8)(5,6)(9,10)(11,21)(12,17)(13,18)(14,19)(15,20)(16,24)(22,25)(23,35)(26,27)(28,34)(29,32)(30,33)(31,37)(36,39)(38,40);(2,6)(7,8)(9,10)(11,22)(12,26)(13,19)(14,18)(15,20)(16,24)(17,27)(21,25)(23,31)(28,33)(29,32)(30,34)(35,37)(36,38)(39,40);(1,3,5,4)(2,7,6,8)(9,11,13,24,15,25,14,29)(10,16,19,21,20,32,18,22)(12,23,27,37,28,39,30,38)(17,31,33,35,34,40,26,36);(1,4,5,3)(2,8,6,7)(9,12)(10,17)(11,23)(13,27)(14,30)(15,28)(16,31)(18,26)(19,33)(20,34)(21,35)(22,36)(24,37)(25,39)(29,38)(32,40);(9,13,15,14)(10,18,20,19)(11,24,25,29)(12,27,28,30)(16,22,32,21)(17,26,34,33)(23,37,39,38)(31,36,40,35)

Polycyclic presentation:

2^9|c^2=fg,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,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, 6351321);