C₂×#3416 #62633

order	256
nilpotency class	4
isoclinism	isoclinic class of #3415
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#3416

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₈	C₂⁵
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ G
LECS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

1037 conjugacy classes of subgroups; only the 479 normal subgroups are shown.

normal subgroups only

This group has 1037 conjugacy classes of subgroups — too many to draw. Only its 479 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 — 5 generators on 34 points:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#3416⋊C₃
maximal subgroup of	#497427, #497428, #497452, …
maximal normal subgroup of	#497427, #497428, #497452, …
maximal quotient of	C₂×#33381, C₂×#33382, C₂×#33411, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55621);