#6465975

order	512
nilpotency class	3
isoclinism	isoclinic class of #6444406
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

1804 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)(11,20)(12,16)(13,18)(14,19)(15,24)(17,29)(21,23)(22,35)(25,27)(26,36)(28,34)(30,37)(31,38)(32,39)(33,40);(2,5)(6,8)(9,10)(11,21)(12,25)(13,17)(14,19)(15,24)(16,27)(18,29)(20,23)(22,32)(26,30)(28,33)(31,38)(34,40)(35,39)(36,37);(9,11,14,23)(10,15)(12,26,28,22)(13,24,29,38)(16,32)(17,20)(18,21)(19,31)(25,35)(27,39,40,37)(30,34)(33,36);(9,12,13,27,14,28,29,40)(10,16,18,25,19,34,17,33)(11,22,24,37,23,26,38,39)(15,30,21,36,31,32,20,35);(1,3,4,7)(2,6,5,8)(11,23)(15,31)(20,21)(22,26)(24,38)(30,32)(35,36)(37,39)

Polycyclic presentation:

2^9|c^2=gh,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,d^c=di,e^c=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, 6373171);