#498275

order	512
nilpotency class	4
isoclinism	isoclinic class of #484039
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×#135	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1770 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

516 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)(4,5)(7,8)(9,10,13,17)(11,19)(12,23,24,37)(14,28,27,18)(15,30)(16,33,34,26)(20,35)(21,32)(22,39)(25,36)(29,38)(31,40);(2,5)(3,7)(11,20,22,29)(12,24)(15,31,32,36)(16,34)(19,35,39,38)(21,25,30,40)(23,37)(26,33);(9,11)(10,15)(12,25)(13,22)(14,29)(16,35)(17,32)(18,36)(19,37)(20,27)(21,33)(23,39)(24,40)(26,30)(28,31)(34,38);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309443, #72309445, #72309455, …
maximal normal subgroup of	#72309443, #72309445, #72309455, …
maximal quotient of	#72343516, #72343605

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405471);