C₂×#7364 #149605

order	512
nilpotency class	4
isoclinism	isoclinic class of #353
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7364

Subgroups

	subgroup	G/H
center	C₂×C₈	#99
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂×M₃₂	C₂³
socle	C₂²	#1322
prime radical	C₂⁴	M₃₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

274 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 64

A minimal faithful permutation representation — 3 generators on 34 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56801);