C₂×#2195 #33537

order	256
nilpotency class	4
isoclinism	isoclinic class of #2192
automorphism	Aut(G) order 8192
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#2195

Subgroups

	subgroup	G/H
center, socle	C₂²	#447
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂×#578	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 539 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) = 66
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 66 points:

(1,2);(3,4,10,16)(5,17,21,45)(6,22,25,42)(7,26,30,40)(8,15,9,14)(11,33,36,19)(12,31,38,24)(13,28,41,29)(18,46,50,63)(20,51,32,39)(23,54,57,37)(27,43,58,35)(34,55,62,66)(44,49,65,48)(47,52,60,61)(53,59,64,56);(3,5)(4,11)(6,18)(7,27)(8,20)(9,32)(10,21)(12,34)(13,39)(14,35)(15,43)(16,36)(17,26)(19,29)(22,52)(23,55)(24,48)(25,50)(28,33)(30,58)(31,49)(37,63)(38,62)(40,45)(41,51)(42,61)(44,53)(46,54)(47,56)(57,66)(59,60)(64,65);(3,6,10,25)(4,12,16,38)(5,18,21,50)(7,23,30,57)(8,31,9,24)(11,34,36,62)(13,37,41,54)(14,42,15,22)(17,44,45,65)(19,47,33,60)(20,49,32,48)(26,53,40,64)(27,55,58,66)(28,59,29,56)(35,61,43,52)(39,63,51,46)

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#2195, #495718, #498165, …
maximal normal subgroup of	C₄×#2195, #495718, #498165, …
maximal quotient of	C₂×#12112, C₂×#12113, #484220, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 26525);