#489182

order	512
nilpotency class	4
isoclinism	isoclinic class of #488901
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

824 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|c^2=h,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fh,g^a=gi,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396378);