C₂×#7239 #127405

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

Aliases

direct product

C₂×#7239

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

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, 34601);