C₂×#12294 #484964

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

Aliases

direct product

C₂×#12294

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3259	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 769 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)(19,20,25,31)(21,32,35,47)(22,26,39,43)(23,30,24,29)(27,40,45,34)(28,38,42,36)(33,46,37,44)(41,49,48,50);(3,5)(4,8)(6,11)(7,12)(9,14)(10,15)(13,18)(16,17)(20,26)(22,36)(23,24)(27,41)(28,30)(29,42)(31,43)(32,46)(33,49)(34,40)(37,50)(38,39)(44,47)(45,48);(3,6,7,13)(4,9,10,16)(5,11,12,18)(8,14,15,17)(19,21,25,35)(20,27,31,45)(22,37,39,33)(23,40,24,34)(26,41,43,48)(28,46,42,44)(29,47,30,32)(36,50,38,49)

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, 392160);