C₂×#7240 #127407

order	512
nilpotency class	3
isoclinism	isoclinic class of #7225
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7240

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,ℤ/4)
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	#405	C₂³
socle	C₂²	#1290
prime radical	C₂⁴	#108
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ #405 ▷ C₂×C₄ ▷ C₂ ▷ C₁

326 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 66
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 66 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34603);