C₂×#12287 #484554

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

Aliases

direct product

C₂×#12287

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3203	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 829 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)(19,20)(21,33)(22,36)(23,39)(24,31)(25,30)(26,32)(27,34)(28,38)(29,42)(35,47)(37,46)(40,49)(41,44)(43,48)(45,50);(3,5)(4,8)(6,11)(7,12)(9,14)(10,15)(13,18)(16,17)(19,21,26,22)(20,27,32,46)(23,40,43,50)(24,35,25,38)(28,44,47,29)(30,45,31,49)(33,48,36,39)(34,41,37,42);(3,6)(4,9)(5,11)(7,13)(8,14)(10,16)(12,18)(15,17)(19,22,26,21)(20,28,32,47)(23,37,43,34)(24,35,25,38)(27,44,46,29)(30,33,31,36)(39,45,48,49)(40,42,50,41)

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, 391750);