#497053

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

549 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)(11,21)(12,25,28,33)(13,29)(14,16)(15,20)(17,26,35,39)(18,23)(19,24)(22,36,38,34)(27,40,37,31)(30,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,14,15,24,23,30)(10,16,18,21,20,32,29,19)(12,26,37,22,28,39,27,38)(17,33,36,31,35,25,34,40);(2,5)(6,8)(9,12,14,26,13,27,24,38,15,28,30,39,23,37,11,22)(10,17,19,25,18,34,16,31,20,35,21,33,29,36,32,40);(1,3,4,7)(2,6,5,8)(9,11,23,30,15,24,13,14)(10,16,29,19,20,32,18,21)(12,22,37,39,28,38,27,26)(17,31,36,25,35,40,34,33)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=f,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	#72313590, #72313605, #72313634, …
maximal normal subgroup of	#72313590, #72313605, #72313634, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404249);