C₂×#7413 #150212

order	512
nilpotency class	5
isoclinism	isoclinic class of #7412
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7413

Subgroups

	subgroup	G/H
center, socle	C₂³	#354
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#391	C₂³
prime radical	C₂⁴	#99
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂×#391 ◁ G
LECS	G ▷ #391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

406 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 82 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)(5,7)(6,9)(8,10)(11,12,14,19,16,21,27,36)(13,22,24,39,23,37,40,35)(15,18,30,34,31,20,28,38)(17,26,32,29,33,41,42,25);(4,7)(5,10)(8,9)(11,13)(12,17,21,33)(14,26)(15,29)(16,23)(18,22,20,37)(19,35,36,39)(24,30)(25,31)(27,41)(28,40)(32,38,42,34)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57408);