C₂×#7306 #128003

order	512
nilpotency class	4
isoclinism	isoclinic class of #7304
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#7306

Subgroups

	subgroup	G/H
center, socle	C₂³	#342
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

1372 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 1372 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 42
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 42 points:

(1,2);(3,4)(5,9)(6,8)(7,10)(11,12,15,20,17,22,16,21)(13,23,25,38,27,40,26,39)(14,28,29,37,30,19,31,35)(18,32,33,24,34,41,36,42);(3,5)(4,7,8,9)(6,10)(11,13,16,26,17,27,15,25)(12,18,21,33,22,34,20,36)(14,24,29,41,30,42,31,32)(19,23,35,39,28,40,37,38)

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35199);