C₈×C₆₁ #92419

order	488
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂²×C₃×C₄×C₅ (order 240) · Inn(G) = C₁ · Out(G) = C₂²×C₃×C₄×C₅
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₄₈₈
abelian	C₈×C₆₁

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄	C₂×C₆₁
socle, prime radical	C₂×C₆₁	C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 488

A minimal faithful permutation representation — 1 generator on 69 points:

(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69)

Conjugacy classes & character table

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

Relationships

center of	M₃₂×C₆₁, #131×C₆₁, C₆₁×#350, …
derived subgroup of	#14519718, D₉₇₆, #14519720, …
Fitting subgroup of	#11758185, C₈×D₆₁, #11758189, …
maximal subgroup of	#11758185, C₁₆×C₆₁, C₈×D₆₁, …
maximal normal subgroup of	#11758185, C₁₆×C₆₁, C₈×D₆₁, …
maximal quotient of	C₁₆×C₆₁, C₂×C₈×C₆₁, C₃⋊C₈×C₆₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(488, 2);

C8×C61 #92419