#354363

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1216 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)(3,4)(5,6,8,7)(9,10)(11,21)(12,26)(13,19)(14,18)(15,20)(16,24)(17,28)(22,30)(23,32)(25,31)(27,34)(29,33)(35,38)(36,40)(37,39);(9,11)(10,16,20,31)(12,22)(13,24)(14,25)(15,23)(17,30,34,40)(18,21,19,32)(26,39,33,35)(27,37)(28,38)(29,36);(2,4)(9,12,13,27,15,29,14,28)(10,17,18,33,20,34,19,26)(11,22,24,37,23,36,25,38)(16,30,21,35,31,40,32,39);(5,7,8,6)(11,23)(16,31)(21,32)(22,36)(24,25)(30,40)(35,39)(37,38)

Polycyclic presentation:

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

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, 261559);