C₂×#13794 #513445

order	512
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂×#13794

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁴
commutator	C₂²	C₂³×C₄²
Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁷	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

19154 conjugacy classes of subgroups; only the 4258 normal subgroups are shown.

normal subgroups only

This group has 19154 conjugacy classes of subgroups — too many to draw. Only its 4258 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) = 128

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

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#13794⋊C₃, #72258367
maximal subgroup of	C₂×#516383, C₂×#5955303, C₂×#513747, …
maximal normal subgroup of	C₂×#516383, C₂×#5955303, C₂×#513747, …
maximal quotient of	C₂×#154037, C₂×#355922, C₂×#153851, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420641);