#206385

order	512
nilpotency class	3
isoclinism	isoclinic class of #19651
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#390
prime radical	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1065 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 113581);