C₂×#13670 #512846

order	512
nilpotency class	5
isoclinism	isoclinic class of #13669
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#13670

Subgroups

	subgroup	G/H
center, socle	C₂²	#2184
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
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

742 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 742 conjugacy classes of subgroups — too many to draw. Only its 146 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)(5,16)(6,20)(7,9)(8,22)(12,34)(13,39)(14,40)(15,33)(17,44)(18,48)(19,26)(21,28)(23,51)(24,50)(25,52)(27,29)(30,32)(31,35)(36,45)(37,59)(38,58)(41,55)(42,46)(47,65)(49,60)(53,57)(54,64)(61,63)(62,66);(3,4)(5,12)(6,21)(7,13)(8,28)(9,30)(10,15)(11,20)(14,22)(16,35)(17,45)(18,49)(19,38)(23,40)(24,55)(25,53)(26,37)(27,41)(29,57)(31,46)(32,50)(33,52)(34,44)(36,43)(39,51)(42,60)(47,66)(48,61)(54,63)(56,62)(58,65)(59,64);(3,5,10,19)(4,12,15,38)(6,17,25,47)(7,26,9,16)(8,18,29,48)(11,31,33,59)(13,37,30,35)(14,36,41,62)(20,46,52,64)(21,45,53,66)(22,43,27,56)(23,54,24,42)(28,49,57,61)(32,58,39,34)(40,63,55,60)(44,50,65,51)

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420042);