#496538

order	512
nilpotency class	4
isoclinism	isoclinic class of #480684
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

956 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)(9,10)(11,15)(12,22,23,32)(13,25,27,17)(14,18)(16,24)(19,33,34,40)(20,36,38,29)(21,30)(26,31)(28,35)(37,39);(1,2)(3,6)(4,5)(7,8)(9,11,14,21)(10,15,18,30)(12,19,23,34)(13,20,27,38)(16,28,31,39)(17,29,25,36)(22,33,32,40)(24,35,26,37);(2,5)(6,8)(9,12,14,23)(10,16,18,31)(11,19,21,34)(13,26,27,24)(15,28,30,39)(17,32,25,22)(20,37,38,35)(29,40,36,33);(1,3,4,7)(2,6,5,8)(12,23)(16,31)(19,34)(22,32)(24,26)(28,39)(33,40)(35,37)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^c=di,f^c=fi,g^c=ghi,g^f=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403734);