C₂×#10345 #237730

order	512
nilpotency class	3
isoclinism	isoclinic class of #9914
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#10345

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1896 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 1896 conjugacy classes of subgroups — too many to draw. Only its 266 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) = 128

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

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 144926);