C₂×#1615 #18106

order	256
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	Aut(G) order 16384
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#1615

Subgroups

	subgroup	G/H
center, socle	C₂³	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
prime radical	C₂×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 508 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) = 34
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 34 points:

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

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#1615, #208871, #208895, …
maximal normal subgroup of	C₄×#1615, #208871, #208895, …
maximal quotient of	C₂×#7787, C₂×#7788, C₂×#7798, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 11094);