#490457

order	512
nilpotency class	4
isoclinism	isoclinic class of #490428
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#33582	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

556 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)(3,6)(7,8)(9,10)(11,21)(12,20)(13,16)(14,27)(15,18)(17,24)(19,29)(22,38)(23,37)(25,32)(26,34)(28,36)(30,35)(31,39)(33,40);(1,2,6,8,4,7,3,5)(10,16)(17,31)(18,20)(19,32)(21,37)(27,34)(33,36)(35,38);(9,11)(10,17)(12,22)(13,23)(14,28)(15,30)(16,31)(18,33)(19,35)(20,36)(21,27)(24,29)(25,39)(26,40)(32,38)(34,37);(9,12,13,15)(10,18,16,20)(11,22,23,30)(14,25,26,29)(17,33,31,36)(19,34,32,27)(21,35,37,38)(24,28,39,40)

Polycyclic presentation:

2^9|b^2=egi,d^2=i,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fh,h^a=hi,f^c=fh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309417, #72309422, #72309432, …
maximal normal subgroup of	#72309417, #72309422, #72309432, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397653);