C₈×#444 #184010

order	512
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₈×#444

Subgroups

	subgroup	G/H
center	C₄×C₈	C₂×D₄
commutator	C₄	C₂²×C₄×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄²×D₄
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

812 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

(1,2,3,5,4,6,7,8);(9,10,13,17,14,11,18,16)(12,19,20,23,21,15,22,24);(9,11)(10,14)(12,19)(13,16)(15,21)(17,18)(20,23)(22,24);(9,12,13,20,14,21,18,22)(10,15,17,24,11,19,16,23)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₈×AGL(1,ℤ/16), C₂×C₈×#444, #72287239, …
maximal normal subgroup of	C₈×AGL(1,ℤ/16), C₂×C₈×#444, #72287239, …
maximal quotient of	C₈×#1580, C₂×C₈×#444

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91206);

C8×#444 #184010

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₈×#444 #184010