#18374

order	256
nilpotency class	3
isoclinism	isoclinic class of #18346
automorphism	Aut(G) order 16384
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
prime radical	C₂³	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

419 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 419 conjugacy classes of subgroups — too many to draw. Only its 115 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) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(17,18,22,26)(19,23,29,36)(20,24,31,38)(21,32,33,25)(27,39,41,34)(28,42,43,35)(30,44,45,37)(40,46,48,47);(1,3,6,13)(2,7,9,15)(4,12,5,11)(8,16,14,10)(17,19)(18,23)(20,27)(21,28)(22,29)(24,34)(25,35)(26,36)(30,40)(31,41)(32,42)(33,43)(37,46)(38,39)(44,47)(45,48);(17,20,22,31)(18,24,26,38)(19,27,29,41)(21,30,33,45)(23,34,36,39)(25,37,32,44)(28,40,43,48)(35,46,42,47);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(17,21,22,33)(18,25,26,32)(19,28,29,43)(20,30,31,45)(23,35,36,42)(24,37,38,44)(27,40,41,48)(34,46,39,47)

Polycyclic presentation:

2^8|a^2=h,b^2=g,c^2=h,d^2=gh,e^2=g,b^a=be,c^a=cf,d^a=dh,e^a=eg,d^b=dg,e^b=eg,f^b=fh,e^c=eh

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#245919, #245926, #245940, …
maximal normal subgroup of	#245919, #245926, #245940, …
maximal quotient of	#168962, #169018, #169074, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 11362);