C₈×#1397 #72234236

order	1024
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₁₆ · Schur multiplier (H₂) = C₂×C₄×C₈

Aliases

direct product

C₈×#1397

Subgroups

	subgroup	G/H
center	C₂×C₈×C₁₆	C₂²
commutator	C₂	C₄×C₈×C₁₆
Frattini, cosocle	C₂×C₄×C₁₆	C₂³
socle, prime radical	C₂³	C₂×C₄×C₁₆
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

661 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 3 generators on 44 points:

(1,2,3,4,5,6,7,8);(13,14,15,20,16,21,23,32,17,22,24,33,26,34,35,42,18,19,25,29,27,30,36,39,28,31,37,40,38,41,43,44);(9,10,11,12)(14,19)(20,29)(21,30)(22,31)(32,39)(33,40)(34,41)(42,44)

Conjugacy classes & character table

640 conjugacy classes. The character table is too large to display.

Relationships

Frattini subgroup of	#72234405
maximal subgroup of	#72234241, #72234246, #72234250, …
maximal normal subgroup of	#72234241, #72234246, #72234250, …
maximal quotient of	#72234297, #72234399

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,2,3,4,5,6,7,8), (13,14,15,20,16,21,23,32,17,22,24,33,26,34,35,42,18,19,25,29,27,30,36,39,28,31,37,40,38,41,43,44), (9,10,11,12)(14,19)(20,29)(21,30)(22,31)(32,39)(33,40)(34,41)(42,44));

C8×#1397 #72234236