C₂×#13702 #512950

order	512
nilpotency class	5
isoclinism	isoclinic class of #13700
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#13702

Subgroups

	subgroup	G/H
center, socle	C₂²	#2215
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂×#2221	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

424 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 138 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);(4,11)(5,18)(6,22)(7,9)(8,23)(12,26)(13,40)(14,34)(15,46)(16,32)(17,35)(19,50)(20,53)(21,27)(24,38)(25,39)(28,29)(30,41)(31,33)(36,45)(37,54)(42,61)(43,59)(44,58)(47,60)(48,51)(52,62)(56,64)(57,63)(65,66);(3,4,12,36,8,15,25,34,7,14,38,46,28,45,6,11,10,17,22,33,29,47,24,32,9,16,39,60,23,31,26,35)(5,13,37,61,20,42,48,40,27,44,50,64,55,65,19,30,21,43,51,57,53,63,54,59,18,41,62,66,49,56,52,58);(3,5)(4,13)(6,19)(7,27)(8,20)(9,18)(10,21)(11,30)(12,37)(14,44)(15,42)(16,41)(17,43)(22,51)(23,49)(24,54)(25,48)(26,52)(28,55)(29,53)(31,56)(32,59)(33,57)(34,40)(35,58)(36,61)(38,50)(39,62)(45,65)(46,64)(47,63)(60,66)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420146);