C₂×#10361 #237890

order	512
nilpotency class	3
isoclinism	isoclinic class of #10078
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#10361

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

994 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 226 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) = 34
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 34 points:

(1,2);(3,4,6,8)(5,9,10,7)(13,18)(15,17)(19,20)(21,23)(22,30)(24,29)(25,27)(26,33)(28,34)(31,32);(3,5,6,10)(4,7,8,9)(11,12)(13,15)(14,16)(17,18)(19,21,23,20)(22,31,32,30)(24,29,25,27)(26,34,28,33);(12,15)(13,18)(16,17)(19,22,23,32)(20,26,21,28)(24,34,25,33)(27,31,29,30)

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, 145086);