C₂×#7366 #149609

order	512
nilpotency class	5
isoclinism	isoclinic class of #1414
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#7366

Subgroups

	subgroup	G/H
center, socle	C₂³	#354
commutator	C₄²	C₂²×C₈
Frattini, cosocle	#382	C₂³
prime radical	C₂⁴	#99
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂×#382 ◁ G
LECS	G ▷ #382 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

346 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 90 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);(3,4,5,7,6,8,9,10)(11,12)(13,21,23,34)(14,25,26,41)(15,28,29,19)(16,20)(17,22,30,37)(18,27)(24,36,40,35)(31,38,32,39)(33,42);(11,13,15,23,16,24,29,40)(12,17,19,31,20,32,28,30)(14,22,27,38,26,39,42,37)(18,21,25,35,33,36,41,34)

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, 56805);