#6465971

order	512
nilpotency class	3
isoclinism	isoclinic class of #6444408
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1826 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)(11,15)(12,17)(13,16)(14,19)(18,27)(20,31)(21,29)(22,30)(23,28)(24,33)(25,34)(26,32)(35,38)(36,40)(37,39);(2,5)(6,8)(9,11)(10,15)(12,24)(13,20)(14,23)(16,31)(17,33)(18,30)(19,28)(21,26)(22,27)(25,36)(29,32)(34,40)(35,39)(37,38);(10,16,19,18)(11,20,23,22)(15,28)(17,32,34,35)(21,36,37,24)(29,39)(30,31)(33,40);(9,12,13,26,14,25,27,38)(10,17,16,32,19,34,18,35)(11,21,22,24,23,37,20,36)(15,29,30,33,28,39,31,40);(1,3,4,7)(2,6,5,8)(12,25)(17,34)(21,37)(24,36)(26,38)(29,39)(32,35)(33,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6373167);