C₂×#13608 #512645

order	512
nilpotency class	5
isoclinism	isoclinic class of #13605
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#13608

Subgroups

	subgroup	G/H
center, socle	C₂²	#2127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	#124	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#536 ◁ G
LECS	G ▷ #124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

800 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 186 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)(5,15,18,42)(6,11)(7,23,25,57)(8,27,29,13)(9,14)(10,17,31,48)(12,26)(16,41,45,63)(19,38,28,43)(20,51,24,56)(21,53,55,36)(22,37)(30,47,61,66)(32,49,33,50)(34,44,62,65)(35,52)(39,58)(40,59)(46,60,54,64);(3,5,9,19)(4,10,14,33)(6,16,22,46)(7,17,25,50)(8,28,29,18)(11,34,37,30)(12,38,40,15)(13,31,27,32)(20,44,24,47)(21,54,55,45)(23,43,57,42)(26,48,59,49)(35,60,39,41)(36,62,53,61)(51,64,56,63)(52,65,58,66);(3,6,9,22)(4,11,14,37)(5,16,19,46)(7,24,25,20)(8,21,29,55)(10,30,33,34)(12,39,40,35)(13,36,27,53)(15,41,38,60)(17,47,50,44)(18,45,28,54)(23,56,57,51)(26,58,59,52)(31,61,32,62)(42,63,43,64)(48,66,49,65)

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419841);