C₂×#2164 #33417

order	256
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	Aut(G) order 16384
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₂×#2164

Subgroups

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

305 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 305 conjugacy classes of subgroups — too many to draw. Only its 207 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) = 42
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 42 points:

(1,2);(3,4)(5,8)(6,7)(9,10)(11,12,14,18,15,19,24,30,16,20,25,31,26,32,36,40)(13,17,22,28,23,29,35,39,21,27,33,37,34,38,41,42);(4,7)(8,10)(11,13)(12,17)(14,22)(15,23)(16,21)(18,28)(19,29)(20,27)(24,35)(25,33)(26,34)(30,39)(31,37)(32,38)(36,41)(40,42);(3,5,6,9)(4,8,7,10)(13,21)(17,27)(22,33)(23,34)(28,37)(29,38)(35,41)(39,42)

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#128284, #128289, #134691, …
maximal normal subgroup of	#128284, #128289, #134691, …
maximal quotient of	C₂×#12592, C₂×#12600, C₂×#12654, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 26405);