#6470073

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470067
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4122 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)(3,5)(4,6)(7,8)(10,16)(11,21)(13,14)(18,31)(19,20)(23,35)(24,29)(25,34)(26,27)(30,33)(37,38)(39,40);(1,3)(2,5)(4,7)(6,8)(9,10,15,20)(11,22,24,17)(12,25,28,18)(13,19,14,16)(21,30,29,33)(23,32,38,36)(26,34,27,31)(35,40,37,39);(3,7)(5,8)(9,11)(10,17)(12,23)(13,21)(14,29)(15,24)(16,30)(18,32)(19,33)(20,22)(25,36)(26,37)(27,35)(28,38)(31,39)(34,40);(3,7)(5,8)(9,12)(10,18)(11,23)(13,27)(14,26)(15,28)(16,31)(17,32)(19,34)(20,25)(21,35)(22,36)(24,38)(29,37)(30,39)(33,40);(2,6)(5,8)(9,13,15,14)(10,19,20,16)(11,21,24,29)(12,26,28,27)(17,33,22,30)(18,31,25,34)(23,37,38,35)(32,39,36,40)

Polycyclic presentation:

2^9|b^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^c=gi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312776, #72312791
maximal normal subgroup of	#72312776, #72312791

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6377269);