#9372

order	256
nilpotency class	3
isoclinism	isoclinic class of #9369
automorphism	Aut(G) = C₂³×#45355 (order 2048) · Inn(G) = #386 · Out(G) = C₂⁵
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#386
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
prime radical	#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

195 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 195 conjugacy classes of subgroups — too many to draw. Only its 59 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) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^8|a^2=f,b^2=df,c^2=g,d^2=gh,e^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=egh,d^b=dh,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fg

Conjugacy classes & character table

37 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#9372, #200168, #200170, …
maximal normal subgroup of	C₂×#9372, #200168, #200170, …
maximal quotient of	#100044, #100050, #100052, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 2360);