#498276

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

508 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)(11,20,23,37)(12,17)(13,25)(14,29)(15,19)(16,22,33,40)(18,28)(21,35,30,38)(24,34)(26,36)(27,32,39,31);(2,5)(3,7)(11,21,23,30)(13,26)(16,31,33,32)(18,34)(20,35,37,38)(22,27,40,39)(24,28)(25,36);(9,11)(10,16)(12,21)(13,27)(14,30)(15,23)(17,32)(18,35)(19,33)(20,36)(22,24)(25,37)(26,39)(28,40)(29,31)(34,38);(1,3,4,7)(2,6,5,8)(9,12,15,14)(10,17,19,29)(11,21,23,30)(13,24,26,28)(16,32,33,31)(18,25,34,36)(20,35,37,38)(22,39,40,27)

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=ei,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	#72309671, #72310191, #72310208, …
maximal normal subgroup of	#72309671, #72310191, #72310208, …
maximal quotient of	#72343355, #72343430, #72343491, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405472);