C₈×M₆₄ #149492

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₁₆ · Schur multiplier (H₂) = C₂×C₈

Aliases

direct product

C₈×M₆₄

Subgroups

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

236 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 40 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);(10,26)(12,28)(14,30)(16,32)(18,34)(20,36)(22,38)(24,40)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	#72210680, C₁₆×M₆₄, #72216220, …
maximal normal subgroup of	#72210680, C₁₆×M₆₄, #72216220, …
maximal quotient of	C₁₆×M₆₄, #72216220, C₂×C₈×M₆₄, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56688);

C8×M64 #149492