C₂×#10346 #237733

order	512
nilpotency class	3
isoclinism	isoclinic class of #9914
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#10346

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

1336 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 1336 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)(14,16,18,17)(19,20)(21,24)(22,25)(23,26)(27,32)(28,31)(29,33)(30,34);(3,5,6,10)(4,7,8,9)(12,15)(16,17)(19,21)(20,24)(22,27)(23,28)(25,32)(26,31)(29,34)(30,33);(12,16)(14,18)(15,17)(19,22,23,29)(20,25,26,33)(21,27,28,34)(24,30,31,32)

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, 144929);