C₈×C₆₄ #94503

order	512
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) order 8192
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₈

Aliases

abelian

C₈×C₆₄

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄×C₃₂	C₂²
socle, prime radical	C₂²	C₄×C₃₂
LCS	G ▷ C₁
UCS	C₁ ◁ G
LECS	G ▷ C₄×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

82 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 2 generators on 72 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);(65,66,67,68,69,70,71,72)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₈×C₁₂₈, C₁₆×C₆₄, C₂×C₈×C₆₄, …
maximal normal subgroup of	C₈×C₁₂₈, C₁₆×C₆₄, C₂×C₈×C₆₄, …
maximal quotient of	C₈×C₁₂₈, C₁₆×C₆₄, C₂×C₈×C₆₄, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1699);

C8×C64 #94503