C₂×#13352 #498896

order	512
nilpotency class	4
isoclinism	isoclinic class of #13350
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#13352

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator, Frattini, cosocle	C₂×#41	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

2089 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 2089 conjugacy classes of subgroups — too many to draw. Only its 194 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)(5,15,19,42)(6,20,23,39)(7,12,26,38)(8,27,29,13)(10,30,33,17)(11,28,35,22)(16,43,46,65)(18,48,50,41)(21,52,24,54)(25,57,58,37)(31,61,62,66)(32,47,59,60)(34,56,36,53)(40,49,64,45)(44,51,55,63);(3,5)(4,10)(6,16)(7,17)(8,18)(9,19)(11,31)(12,15)(13,32)(14,33)(20,51)(21,44)(22,45)(23,46)(24,55)(25,47)(26,30)(27,59)(28,49)(29,50)(34,40)(35,62)(36,64)(37,41)(38,42)(39,63)(43,52)(48,57)(53,66)(54,65)(56,61)(58,60);(3,6,9,23)(4,11,14,35)(5,16,19,46)(7,24,26,21)(8,28,29,22)(10,31,33,62)(12,36,38,34)(13,39,27,20)(15,40,42,64)(17,44,30,55)(18,49,50,45)(25,53,58,56)(32,63,59,51)(37,52,57,54)(41,65,48,43)(47,61,60,66)

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, 406092);