#488973

order	512
nilpotency class	4
isoclinism	isoclinic class of #488916
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#61088	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

915 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396169);