C₂×#10586 #241235

order	512
nilpotency class	3
isoclinism	isoclinic class of #10078
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#10586

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

858 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 858 conjugacy classes of subgroups — too many to draw. Only its 242 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) = 512

A minimal faithful permutation representation — 4 generators on 42 points:

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

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, 148431);