#10045

order	256
nilpotency class	3
isoclinism	isoclinic class of #10038
automorphism	Aut(G) = C₂×#72278822 (order 4096) · Inn(G) = #392 · Out(G) = C₂×#120
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂³	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

222 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 49 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)(3,9)(4,12)(5,8)(6,11)(7,14)(10,16)(13,15)(17,18,23,28)(19,29,32,40)(20,25,36,43)(21,27,22,26)(24,33,39,31)(30,45,46,41)(34,48,47,38)(35,44,37,42);(6,15)(9,16)(17,19,23,32)(18,24,28,39)(20,34,36,47)(21,33,22,31)(25,41,43,45)(26,40,27,29)(30,37,46,35)(38,44,48,42);(1,3)(2,6)(4,13)(5,10)(7,16)(8,15)(9,12)(11,14)(17,20,23,36)(18,25,28,43)(19,30,32,46)(21,35,22,37)(24,38,39,48)(26,42,27,44)(29,41,40,45)(31,34,33,47)

Polycyclic presentation:

2^8|a^2=g,b^2=g,c^2=g,d^2=g,e^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgh,c^b=cf,d^b=dg,e^b=eg,f^b=fg,d^c=dh,e^c=eh,f^c=fg

Conjugacy classes & character table

31 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#10045, #235539, #235541, …
maximal normal subgroup of	C₂×#10045, #235539, #235541, …
maximal quotient of	#102474, #102478, #102485, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 3033);