#503529

order	512
nilpotency class	4
isoclinism	isoclinic class of #501072
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33190	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

810 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:

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

Polycyclic presentation:

2^9|a^2=ehi,b^2=e,c^2=ef,d^2=ei,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=cg,g^b=gi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410725);