C₂×#9979 #232439

order	512
nilpotency class	3
isoclinism	isoclinic class of #9974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#9979

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1350 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1350 conjugacy classes of subgroups — too many to draw. Only its 210 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)(19,20,25,29)(21,30,34,42)(22,35,39,45)(23,40,24,28)(26,33,43,32)(27,37,46,38)(31,48,49,41)(36,47,50,44);(3,5,7,12)(4,8,10,15)(6,11,13,18)(9,16,17,14)(19,21)(20,26)(22,36)(23,33)(24,32)(25,34)(27,44)(28,42)(29,43)(30,40)(31,37)(35,41)(38,49)(39,50)(45,48)(46,47);(3,6)(4,9)(5,11)(7,13)(8,14)(10,17)(12,18)(15,16)(19,22,25,39)(20,27,29,46)(21,31,34,49)(23,37,24,38)(26,41,43,48)(28,45,40,35)(30,47,42,44)(32,36,33,50)

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139635);