C₂×#9985 #232511

order	512
nilpotency class	3
isoclinism	isoclinic class of #9980
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#9985

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1374 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1374 conjugacy classes of subgroups — too many to draw. Only its 202 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) = 128

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

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139707);