#202998

order	512
nilpotency class	3
isoclinism	isoclinic class of #19162
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#388
prime radical	C₂⁴	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

723 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,d^c=dh,f^c=fhi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 110194);