#497145

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	#3147	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

659 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)(3,6)(4,7)(5,8)(9,10,14,18)(11,19,23,35)(12,24)(13,28,20,17)(15,29,30,22)(16,31)(21,34)(25,33)(26,39)(27,37)(32,36)(38,40);(1,3,6,2)(4,7,5,8)(9,11,20,29,14,23,13,22)(10,15,28,35,18,30,17,19)(12,25,26,31,27,38,36,21)(16,32,33,37,34,39,40,24);(1,4,6,5)(2,7,3,8)(9,12,23,31,13,26,29,38,14,27,11,21,20,36,22,25)(10,16,30,37,17,33,35,39,18,34,15,24,28,40,19,32);(1,5,6,4)(2,7,3,8)(9,11,13,22,14,23,20,29)(10,15,17,19,18,30,28,35)(12,21,26,25,27,31,36,38)(16,24,33,32,34,37,40,39)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=efh,d^2=e,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	#72309410, #72309429, #72309431, …
maximal normal subgroup of	#72309410, #72309429, #72309431, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404341);