#6467256

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442587
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 ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1923 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)(3,6)(4,5)(7,8)(9,10)(11,21)(12,25)(13,17)(14,19)(15,31)(16,20)(18,29)(22,38)(23,33)(24,34)(26,32)(27,40)(28,37)(30,36)(35,39);(9,11,16,23)(10,17,20,34)(12,26,29,30)(13,15,24,14)(18,35,25,38)(19,33,31,21)(22,28,39,27)(32,40,36,37);(2,5)(6,8)(9,12,14,27,16,29,15,28)(10,18,19,37,20,25,31,40)(11,22,24,26,23,39,13,30)(17,32,21,35,34,36,33,38);(9,13,14,11,16,24,15,23)(10,17,19,21,20,34,31,33)(12,26,27,39,29,30,28,22)(18,36,37,38,25,32,40,35);(1,3,4,7)(2,6,5,8)(9,14,16,15)(10,19,20,31)(11,13,23,24)(12,27,29,28)(17,33,34,21)(18,37,25,40)(22,30,39,26)(32,38,36,35)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320791
maximal normal subgroup of	#72320791

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374452);