C₂×#13612 #512673

order	512
nilpotency class	5
isoclinism	isoclinic class of #13609
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#13612

Subgroups

	subgroup	G/H
center, socle	C₂²	#2127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₄⋊Q₈	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 ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 792 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);(3,4,7,13,9,15,27,42)(5,16,18,40,20,33,47,59)(6,11,23,36,22,37,51,64)(8,14,29,41,30,12,26,39)(10,25,32,28,34,46,60,19)(17,43,44,62,21,48,49,65)(24,35,55,61,52,38,54,63)(31,56,57,50,58,53,66,45);(3,5,9,20)(4,10,15,34)(6,21,22,17)(7,25,27,46)(8,28,30,19)(11,31,37,58)(12,33,14,16)(13,40,42,59)(18,26,47,29)(23,53,51,56)(24,45,52,50)(32,41,60,39)(35,43,38,48)(36,62,64,65)(44,55,49,54)(57,61,66,63);(3,6)(4,11)(5,17)(7,23)(8,24)(9,22)(10,31)(12,38)(13,36)(14,35)(15,37)(16,43)(18,44)(19,45)(20,21)(25,56)(26,54)(27,51)(28,50)(29,55)(30,52)(32,57)(33,48)(34,58)(39,63)(40,62)(41,61)(42,64)(46,53)(47,49)(59,65)(60,66)

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, 419869);