#491797

order	512
nilpotency class	4
isoclinism	isoclinic class of #491728
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×SD₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

480 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)(3,8)(6,7)(9,10)(11,16)(12,18)(13,17)(14,28)(15,20)(19,26)(21,30)(22,29)(23,38)(24,32)(25,33)(27,34)(31,36)(35,39)(37,40);(1,2,4,7)(3,6,8,5)(9,11,15,24)(10,16,20,32)(12,21,14,23)(13,22,27,37)(17,29,34,40)(18,30,28,38)(19,31,33,39)(25,35,26,36);(10,17)(12,14)(13,26)(16,29)(18,19)(20,34)(21,23)(22,36)(25,27)(28,33)(30,31)(32,40)(35,37)(38,39);(2,6)(3,8)(5,7)(9,12,15,14)(10,18,20,28)(11,21,24,23)(13,25,27,26)(16,30,32,38)(17,33,34,19)(22,35,37,36)(29,39,40,31)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309937, #72309991
maximal normal subgroup of	#72309937, #72309991

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398993);