#6467259

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442584
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×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1912 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)(3,6)(4,5)(7,8)(9,10)(11,22)(12,18)(13,19)(14,31)(15,20)(16,21)(17,25)(23,33)(24,37)(26,34)(27,38)(28,35)(29,36)(30,40)(32,39);(9,11)(10,17)(12,24)(13,30)(14,26)(15,25)(16,27)(18,33)(19,37)(20,38)(21,34)(22,31)(23,29)(28,39)(32,36)(35,40);(2,5)(6,8)(9,12,15,29,16,28,14,13)(10,18,20,36,21,35,31,19)(11,23,26,24,27,30,25,39)(17,32,22,33,34,37,38,40);(9,13,14,28,16,29,15,12)(10,19,31,35,21,36,20,18)(11,24,25,23,27,39,26,30)(17,33,38,32,34,40,22,37);(1,3,4,7)(2,6,5,8)(12,28)(13,29)(18,35)(19,36)(23,30)(24,39)(32,37)(33,40)

Polycyclic presentation:

2^9|c^2=h,d^2=hi,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,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320582, #72320791
maximal normal subgroup of	#72320582, #72320791
maximal quotient of	#72321441, #72321621, #72321742, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374455);