#497559

order	512
nilpotency class	4
isoclinism	isoclinic class of #481415
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3341	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

836 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)(6,8)(9,10,11,15,14,19,22,18)(12,23,25,34,26,39,20,36)(13,27,21,17,29,31,38,35)(16,28,32,24,33,40,30,37);(1,2)(3,6)(4,5)(7,8)(9,11,14,22)(10,15,19,18)(12,20,26,25)(13,21,29,38)(16,30,33,32)(17,31,35,27)(23,36,39,34)(24,28,37,40);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,20)(13,28)(14,26)(15,30)(17,34)(18,32)(19,33)(21,37)(22,25)(23,31)(24,38)(27,39)(29,40)(35,36);(2,5)(6,8)(9,11,14,22)(10,15,19,18)(12,20,26,25)(13,21,29,38)(16,30,33,32)(17,31,35,27)(23,36,39,34)(24,28,37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313688, #72313701, #72313707, …
maximal normal subgroup of	#72313688, #72313701, #72313707, …
maximal quotient of	#72343323, #72343396

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404755);