Q₈×C₆₄ #153599

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂²×C₆₄ · Schur multiplier (H₂) = C₂²

Aliases

direct product

Q₈×C₆₄

Subgroups

	subgroup	G/H
center	C₂×C₆₄	C₂²
commutator	C₂	C₂²×C₆₄
Frattini, cosocle	C₂×C₃₂	C₂³
socle, prime radical	C₂²	C₂²×C₃₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₄ ◁ G
LECS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

99 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2,3,4)(5,8,7,6);(1,5,3,7)(2,6,4,8);(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

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

Relationships

maximal subgroup of	SL(2,𝔽₃)×C₆₄, C₃×Q₈×C₆₄
maximal normal subgroup of	SL(2,𝔽₃)×C₆₄, C₃×Q₈×C₆₄
maximal quotient of	C₃×Q₈×C₆₄

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60795);

Q8×C64 #153599