#496883

order	512
nilpotency class	4
isoclinism	isoclinic class of #483928
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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	C₂×#466	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

555 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)(4,5)(7,8)(10,16)(11,20,23,40)(12,15)(14,27)(17,32,31,33)(19,36)(21,37,34,38)(22,29,30,39)(25,26)(28,35);(2,5)(3,7)(9,10)(11,21,23,34)(12,24)(13,16)(14,28)(15,18)(17,30,31,22)(19,26)(20,38,40,37)(25,35)(27,36)(29,33,39,32);(1,3,4,7)(2,6,5,8)(9,11,13,23)(10,17,16,31)(12,22,15,30)(14,29,26,39)(18,21,24,34)(19,37,28,38)(20,27,40,25)(32,36,33,35);(2,5)(6,8)(9,12,13,15)(10,18,16,24)(11,22,23,30)(14,25,26,27)(17,21,31,34)(19,35,28,36)(20,39,40,29)(32,38,33,37)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264915, #72264923, #72264946, …
maximal normal subgroup of	#72264915, #72264923, #72264946, …
maximal quotient of	#72320433, C₂×#496883, #72343728

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404079);