C₂×#12296 #484965

order	512
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#12296

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

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

normal subgroups only

This group has 705 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) = 50
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 50 points:

(1,2);(3,4,7,10)(5,8,12,15)(6,9,13,16)(11,17,18,14)(20,26)(21,30)(22,33)(23,24)(27,41)(28,45)(29,40)(31,47)(32,37)(34,38)(35,36)(39,42)(43,50)(44,48);(3,5)(4,8)(6,11)(7,12)(9,14)(10,15)(13,18)(16,17)(19,20,25,29)(21,27,32,44)(22,34,36,45)(23,28,24,38)(26,35,40,33)(30,42,37,43)(31,48,47,41)(39,49,50,46);(3,6,7,13)(4,9,10,16)(5,11,12,18)(8,14,15,17)(19,21)(20,27)(22,31)(23,37)(24,30)(25,32)(26,39)(28,43)(29,44)(33,46)(34,48)(35,49)(36,47)(38,42)(40,50)(41,45)

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, 392161);