#265209

order	512
nilpotency class	3
isoclinism	isoclinic class of #265163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18766	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

928 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)(3,4)(5,7)(6,8)(10,16)(12,24)(13,14)(17,20)(18,25)(19,28)(21,35)(22,29)(26,27)(30,33)(34,36)(37,40);(1,3,6,7)(2,4,8,5)(9,10,14,28,15,19,13,16)(11,20,29,33,23,30,22,17)(12,25,26,31,27,18,24,38)(21,36,40,32,37,34,35,39);(2,3)(4,5)(7,8)(9,11)(10,17)(12,21)(13,22)(14,29)(15,23)(16,30)(18,32)(19,33)(20,28)(24,35)(25,39)(26,40)(27,37)(31,36)(34,38);(1,4,6,5)(2,3,8,7)(9,12,14,26,15,27,13,24)(10,18,28,38,19,25,16,31)(11,21,29,40,23,37,22,35)(17,32,20,34,33,39,30,36)

Polycyclic presentation:

2^9|b^2=ehi,d^2=ehi,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,f^b=fhi,g^b=ghi,d^c=di,f^c=fi,g^c=ghi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 172405);