C₂×#3549 #62757

order	256
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂×#3549

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁶
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂² ▷ C₁

4401 conjugacy classes of subgroups; only the 2863 normal subgroups are shown.

normal subgroups only

This group has 4401 conjugacy classes of subgroups — too many to draw. Only its 2863 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) = 18
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 18 points:

(1,2);(4,6)(8,9)(12,15)(13,14);(3,4,7,6)(5,9,10,8)(11,12,16,15)(13,18,14,17);(11,13,16,14)(12,17,15,18);(3,5,7,10)(4,8,6,9)(11,14,16,13)(12,18,15,17);(3,6,7,4)(5,9,10,8)(11,15,16,12)(13,18,14,17)

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#3549⋊C₃, C₂×#11714278, C₂×#11715758, …
maximal subgroup of	#185839, #186762, #186776, …
maximal normal subgroup of	#185839, #186762, #186776, …
maximal quotient of	C₂×#34368, C₂×#34415, C₂×#34485, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55745);