#253369

order	512
nilpotency class	3
isoclinism	isoclinic class of #253363
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18332	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

880 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=eh,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,f^b=fi,e^c=eh,f^c=fi,g^c=gh,f^d=fh

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, 160565);