#489069

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	#33582	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

721 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)(3,7)(6,8)(10,17)(11,22)(12,15)(13,27)(14,26)(19,28)(20,30)(21,33)(23,29)(24,31)(25,40)(32,36)(34,37)(35,39);(1,2,4,6)(3,8,7,5)(9,10)(11,18)(12,20)(13,28)(14,30)(15,19)(16,21)(17,27)(22,39)(23,37)(24,34)(25,38)(26,33)(29,36)(31,35)(32,40);(9,11)(10,18)(12,23)(13,29)(14,31)(15,24)(16,25)(17,32)(19,34)(20,37)(21,38)(22,26)(27,40)(28,36)(30,35)(33,39);(9,12,16,15)(10,19,21,20)(11,23,25,24)(13,26,14,27)(17,30,33,28)(18,34,38,37)(22,31,40,29)(32,35,39,36)

Polycyclic presentation:

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

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, 396265);