C₂×#12305 #484980

order	512
nilpotency class	4
isoclinism	isoclinic class of #12304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#12305

Subgroups

	subgroup	G/H
center, socle	C₂²	#1616
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 664 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);(3,4,8,28,9,14,30,13)(5,15,19,37,18,42,29,43)(6,20,23,36,22,11,34,50)(7,24,27,12,26,56,57,39)(10,25,40,17,33,47,32,48)(16,41,45,65,46,59,55,62)(21,51,53,38,54,63,61,35)(31,58,64,66,49,52,60,44);(3,5,9,18)(4,10,14,33)(6,16,22,46)(7,25,26,47)(8,29,30,19)(11,31,20,49)(12,37,39,43)(13,40,28,32)(15,24,42,56)(17,27,48,57)(21,52,54,58)(23,55,34,45)(35,62,38,65)(36,64,50,60)(41,51,59,63)(44,53,66,61);(3,6)(4,11,14,20)(5,16)(7,21)(8,23)(9,22)(10,31,33,49)(12,38,39,35)(13,36,28,50)(15,41,42,59)(17,44)(18,46)(19,45)(24,51,56,63)(25,52)(26,54)(27,53)(29,55)(30,34)(32,60,40,64)(37,65,43,62)(47,58)(48,66)(57,61)

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, 392176);