#6470427

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470421
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2680 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)(3,5)(4,6)(7,8)(10,15)(11,20)(13,16)(18,30)(19,31)(22,35)(23,24)(25,34)(26,27)(29,33)(37,38)(39,40);(1,3)(2,5)(4,7)(6,8)(9,10,16,31,14,19,13,15)(11,21,20,29,24,17,23,33)(12,25,26,30,28,18,27,34)(22,32,35,40,38,36,37,39);(3,7)(5,8)(9,11)(10,17)(12,22)(13,23)(14,24)(15,29)(16,20)(18,32)(19,21)(25,36)(26,35)(27,37)(28,38)(30,39)(31,33)(34,40);(3,7)(5,8)(9,12,14,28)(10,18,19,25)(11,22,24,38)(13,27,16,26)(15,30,31,34)(17,32,21,36)(20,35,23,37)(29,39,33,40);(2,6)(5,8)(9,13,14,16)(10,15,19,31)(11,23,24,20)(12,26,28,27)(17,29,21,33)(18,34,25,30)(22,35,38,37)(32,40,36,39)

Polycyclic presentation:

2^9|b^2=fi,d^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,g^c=gi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6377623);