C₂×#12168 #484633

order	512
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#12168

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 867 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) = 74
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 74 points:

(1,2);(7,10)(11,12,18,24)(13,25,29,53)(14,30,33,50)(15,34,38,48)(16,23,17,22)(19,41,44,27)(20,39,46,32)(21,36,49,37)(26,54,58,71)(28,59,40,47)(31,62,65,45)(35,51,66,43)(42,63,70,74)(52,57,73,56)(55,60,68,69)(61,67,72,64);(3,4)(5,7)(6,8)(9,10)(11,13,18,29)(12,19,24,44)(14,26,33,58)(15,35,38,66)(16,28,17,40)(20,42,46,70)(21,47,49,59)(22,43,23,51)(25,48,53,34)(27,36,41,37)(30,60,50,69)(31,63,65,74)(32,56,39,57)(45,71,62,54)(52,72,73,61)(55,67,68,64);(3,5,6,9)(4,7,8,10)(11,14)(12,20)(13,26)(15,31)(16,39)(17,32)(18,33)(19,42)(21,45)(22,50)(23,30)(24,46)(25,52)(27,55)(28,57)(29,58)(34,61)(35,63)(36,67)(37,64)(38,65)(40,56)(41,68)(43,69)(44,70)(47,71)(48,72)(49,62)(51,60)(53,73)(54,59)(66,74)

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391829);