C₂×#1287 #7963

order	256
nilpotency class	3
isoclinism	isoclinic class of #1284
automorphism	Aut(G) order 16384
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₂×#1287

Subgroups

	subgroup	G/H
center, socle	C₂³	#95
commutator	C₂×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂⁴	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

273 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 273 conjugacy classes of subgroups — too many to draw. Only its 127 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 — 3 generators on 34 points:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#1287, #105397, #105538, …
maximal normal subgroup of	C₄×#1287, #105397, #105538, …
maximal quotient of	C₂×#7018, #94999, #95005, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 951);