C₂×#2101 #23742

order	256
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	Aut(G) order 131072
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#2101

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

455 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 455 conjugacy classes of subgroups — too many to draw. Only its 199 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) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 26 points:

(1,2);(3,4,7,9)(5,10,6,8)(11,12)(13,15)(14,16)(17,18)(19,20,22,23)(21,24,26,25);(3,5,7,6)(4,8,9,10)(11,13,14,17)(12,15,16,18)(19,21,22,26)(20,24,23,25);(3,6,7,5)(4,8,9,10)(12,16)(15,18)(20,23)(21,25)(24,26)

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#2101, #359367, #359380, …
maximal normal subgroup of	C₄×#2101, #359367, #359380, …
maximal quotient of	C₂×#10689, #244082, #244099, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 16730);