C₂×#13431 #499118

order	512
nilpotency class	4
isoclinism	isoclinic class of #13429
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#13431

Subgroups

	subgroup	G/H
center, socle	C₂²	#2044
commutator, Frattini, cosocle	C₂×#41	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1065 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 1065 conjugacy classes of subgroups — too many to draw. Only its 178 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 — 4 generators on 66 points:

(1,2);(3,4,9,14)(5,15,19,38)(6,20,24,41)(7,25,28,12)(8,13,31,42)(10,26,34,17)(11,30,37,23)(16,45,21,53)(18,44,29,62)(22,55,57,52)(27,51,58,39)(32,63,35,65)(33,60,40,48)(36,56,59,61)(43,49,66,47)(46,64,54,50);(3,5,9,19)(4,10,14,34)(6,21,24,16)(7,26,28,17)(8,29,31,18)(11,35,37,32)(12,38,25,15)(13,40,42,33)(20,50,41,64)(22,46,57,54)(23,49,30,47)(27,48,58,60)(36,43,59,66)(39,44,51,62)(45,55,53,52)(56,65,61,63);(3,6,7,22,9,24,28,57)(4,11,12,36,14,37,25,59)(5,16,17,46,19,21,26,54)(8,30,27,61,31,23,58,56)(10,32,15,43,34,35,38,66)(13,41,39,55,42,20,51,52)(18,49,48,63,29,47,60,65)(33,50,44,53,40,64,62,45)

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, 406314);