#204429

order	512
nilpotency class	3
isoclinism	isoclinic class of #471
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#121
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#390
prime radical	C₂²×#135	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1208 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=eh,g^b=gh

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 111625);