#497147

order	512
nilpotency class	4
isoclinism	isoclinic class of #483987
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33719	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

675 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264744, #72264781, #72265287, …
maximal normal subgroup of	#72264744, #72264781, #72265287, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404343);