#496526

order	512
nilpotency class	4
isoclinism	isoclinic class of #480662
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

938 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)(9,10,13,18,14,17,25,31)(11,15,20,28,21,27,36,39)(12,22,24,32,23,16,29,30)(19,33,35,40,34,26,37,38);(1,2)(3,6)(4,5)(7,8)(9,11,14,21)(10,15,17,27)(12,19,23,34)(13,20,25,36)(16,26,22,33)(18,28,31,39)(24,35,29,37)(30,38,32,40);(2,5)(6,8)(9,12,14,23)(10,16,17,22)(11,19,21,34)(13,24,25,29)(15,26,27,33)(18,32,31,30)(20,35,36,37)(28,40,39,38);(1,3,4,7)(2,6,5,8)(12,23)(16,22)(19,34)(24,29)(26,33)(30,32)(35,37)(38,40)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^c=di,g^c=gh,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, 403722);