#6470047

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470037
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

4090 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)(13,14)(17,32)(18,31)(19,28)(20,30)(22,29)(24,34)(25,27)(33,40)(35,39)(36,37);(1,3)(2,5)(4,7)(6,8)(9,10)(11,20)(12,24)(13,16)(14,28)(15,19)(17,23)(18,26)(21,33)(22,32)(25,31)(27,34)(29,30)(35,38)(36,40)(37,39);(3,7)(5,8)(9,11)(10,17)(12,21)(13,22)(14,29)(15,23)(16,30)(18,33)(19,20)(24,35)(25,36)(26,38)(27,37)(28,32)(31,39)(34,40);(3,7)(5,8)(9,12,15,26)(10,18,19,24)(11,21,23,38)(13,27,14,25)(16,31,28,34)(17,33,20,35)(22,37,29,36)(30,39,32,40);(2,6)(5,8)(9,13,15,14)(10,16,19,28)(11,22,23,29)(12,25,26,27)(17,30,20,32)(18,34,24,31)(21,36,38,37)(33,40,35,39)

Polycyclic presentation:

2^9|d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,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, 6377243);