#354366

order	512
nilpotency class	3
isoclinism	isoclinic class of #3055
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×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1224 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)(17,29)(22,30)(23,32)(25,31)(27,34)(28,33)(35,37)(36,39)(38,40);(9,11)(10,16,20,31)(12,22)(13,24)(14,25)(15,23)(17,30,33,40)(18,21,19,32)(26,39,34,35)(27,38)(28,36)(29,37);(2,4)(9,12,13,28,15,27,14,29)(10,17,18,34,20,33,19,26)(11,22,24,36,23,38,25,37)(16,30,21,35,31,40,32,39);(5,7,8,6)(11,23)(12,27)(16,31)(17,33)(21,32)(24,25)(26,34)(28,29)

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,d^c=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, 261562);