C₂×#13749 #513327

order	512
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄³ · Schur multiplier (H₂) = C₂⁶×C₄³

Aliases

direct product

C₂×#13749

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁴
commutator	C₂	C₂²×C₄³
Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

8970 conjugacy classes of subgroups; only the 4354 normal subgroups are shown.

normal subgroups only

This group has 8970 conjugacy classes of subgroups — too many to draw. Only its 4354 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) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 22 points:

(1,2);(7,8,10,9)(15,16,19,17)(18,21,22,20);(11,12,14,13)(16,17)(20,21);(3,4,5,6)(7,9,10,8)(11,13,14,12)(15,17,19,16)(18,21,22,20);(15,18)(16,20)(17,21)(19,22)

Conjugacy classes & character table

272 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup of	C₂×#13749⋊C₃, #72268650
maximal subgroup of	C₂×#110561, C₂×#110998, C₂×#110560, …
maximal normal subgroup of	C₂×#110561, C₂×#110998, C₂×#110560, …
maximal quotient of	C₂×#153709, C₂×C₂²⋊C₄×#95, C₂²×#13749

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420523);