C₂×#1617 #18118

order	256
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	Aut(G) order 32768
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#1617

Subgroups

	subgroup	G/H
center, socle	C₂³	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
prime radical	C₂⁴	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

492 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 135 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 — 4 generators on 34 points:

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

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#137326, #137328, #137345, …
maximal normal subgroup of	#137326, #137328, #137345, …
maximal quotient of	C₂×#7792, C₂×#7794, C₂×#7827, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 11106);