C₄×M₁₂₈ #153422

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₃₂ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₄×M₁₂₈

Subgroups

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

152 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 68 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);(6,38)(8,40)(10,42)(12,44)(14,46)(16,48)(18,50)(20,52)(22,54)(24,56)(26,58)(28,60)(30,62)(32,64)(34,66)(36,68)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	#72234363, C₈×M₁₂₈, #72234367, …
maximal normal subgroup of	#72234363, C₈×M₁₂₈, #72234367, …
maximal quotient of	C₈×M₁₂₈

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60618);

C4×M128 #153422