C₂×#1340 #11684

order	256
nilpotency class	4
isoclinism	isoclinic class of #1339
automorphism	Aut(G) order 4096
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#1340

Subgroups

	subgroup	G/H
center, socle	C₂²	#327
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂×M₁₆	C₂³
prime radical	C₂⁴	D₈
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 222 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)(5,14,16,31)(6,19,21,30)(7,12)(8,24)(9,13)(10,15)(11,22,28,23)(17,27,18,34)(20,29)(25,32)(26,33);(3,5,9,16)(4,10,13,26)(6,20,21,32)(7,18,8,17)(11,27,28,34)(12,29,24,25)(14,30,31,19)(15,22,33,23)

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#1340, #128805, #128833, …
maximal normal subgroup of	C₄×#1340, #128805, #128833, …
maximal quotient of	C₂×#7078, C₂×#7079, #122539, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 4672);