#263550

order	512
nilpotency class	3
isoclinism	isoclinic class of #263527
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	#19053	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

855 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)(3,5)(4,6)(7,8)(9,10,15,19)(11,20,25,22)(12,26,30,17)(13,32,18,14)(16,24,35,23)(21,36,37,38)(27,34,39,40)(28,31,33,29);(2,5)(4,8)(9,11,14,23,15,25,32,24)(10,16,18,20,19,35,13,22)(12,27,29,37,30,39,31,21)(17,36,28,40,26,38,33,34);(2,6)(4,8)(5,7)(9,12)(10,17)(11,21)(13,33)(14,29)(15,30)(16,34)(18,28)(19,26)(20,36)(22,38)(23,27)(24,39)(25,37)(31,32)(35,40);(2,5)(6,7)(9,13,14,10,15,18,32,19)(11,22,23,16,25,20,24,35)(12,28,29,26,30,33,31,17)(21,36,27,40,37,38,39,34)

Polycyclic presentation:

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

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, 170746);