C₂×#3027 #60396

order	256
nilpotency class	3
isoclinism	isoclinic class of #3026
automorphism	Aut(G) order 196608
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

direct product

C₂×#3027

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#120
commutator, Frattini, cosocle	C₂³	C₂⁵
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₂ ▷ C₁

2555 conjugacy classes of subgroups; only the 559 normal subgroups are shown.

normal subgroups only

This group has 2555 conjugacy classes of subgroups — too many to draw. Only its 559 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) = 32

A minimal faithful permutation representation — 5 generators on 18 points:

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

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#3027⋊C₃, #72228017, C₂×#11714184, …
maximal subgroup of	#147753, #147762, #147764, …
maximal normal subgroup of	#147753, #147762, #147764, …
maximal quotient of	C₂×#18300, C₂×#18301, C₂×#18302, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53384);