C₂×#13438 #499124

order	512
nilpotency class	4
isoclinism	isoclinic class of #13436
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#13438

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

1073 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 1073 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);(3,4,10,15)(5,16,21,41)(6,22,27,55)(7,28,9,13)(8,30,32,14)(11,20,36,18)(12,37,40,25)(17,46,49,66)(19,51,31,45)(23,56,47,62)(24,54,26,53)(29,42,33,44)(34,57,63,64)(35,50,43,52)(38,61,65,48)(39,58,60,59);(3,5,10,21)(4,11,15,36)(6,23,27,47)(7,20,9,18)(8,31,32,19)(12,38,40,65)(13,41,28,16)(14,43,30,35)(17,24,49,26)(22,46,55,66)(25,57,37,64)(29,50,33,52)(34,60,63,39)(42,45,44,51)(48,58,61,59)(53,62,54,56);(3,6,7,24,10,27,9,26)(4,12,13,39,15,40,28,60)(5,17,18,47,21,49,20,23)(8,25,29,58,32,37,33,59)(11,34,16,38,36,63,41,65)(14,22,42,53,30,55,44,54)(19,48,50,64,31,61,52,57)(35,62,45,46,43,56,51,66)

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, 406320);