C₂×#1419 #13623

order	256
nilpotency class	5
isoclinism	isoclinic class of #1418
automorphism	Aut(G) order 8192
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#1419

Subgroups

	subgroup	G/H
center, socle	C₂²	#357
commutator	C₁₆	C₂²×C₄
Frattini, cosocle	C₂×C₁₆	C₂³
prime radical	C₂²×D₁₆	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

163 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 163 conjugacy classes of subgroups — too many to draw. Only its 63 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);(3,4,6,12,9,14,8,25)(5,15,18,32,21,10,20,26)(7,22,17,29,24,30,23,13)(11,28,27,16,31,34,33,19);(4,10)(5,16)(6,8)(7,23)(11,29)(12,32)(13,31)(14,15)(17,24)(18,28)(19,21)(20,34)(22,27)(25,26)(30,33)

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#1419, #151969, #152162, …
maximal normal subgroup of	C₄×#1419, #151969, #152162, …
maximal quotient of	C₂×#7385, #150064, #150124, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6611);