#8501

order	256
nilpotency class	3
isoclinism	isoclinic class of #8491
automorphism	Aut(G) = C₂²×#648360 (order 2048) · Inn(G) = #379 · Out(G) = C₂²×D₄
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#379
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
prime radical	#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

308 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 69 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,5,12,8,3,14,18)(4,20,23,53,25,45,55,52)(6,11,27,39,31,17,42,49)(7,13,28,10,32,19,43,16)(9,33,35,62,15,44,34,29)(21,54,24,38,26,59,58,48)(22,36,56,41,57,46,61,51)(30,40,60,37,63,50,64,47);(1,3)(2,8)(4,15)(5,10)(6,11)(7,19)(9,25)(12,28)(13,32)(14,16)(17,31)(18,43)(20,22)(21,34)(23,41)(24,46)(26,35)(27,47)(29,38)(30,40)(33,52)(36,58)(37,42)(39,64)(44,53)(45,57)(48,62)(49,60)(50,63)(51,55)(54,56)(59,61);(1,4,7,24)(2,9,13,36)(3,15,19,46)(5,26,28,23)(6,29,30,22)(8,25,32,58)(10,35,12,41)(11,38,40,20)(14,21,43,55)(16,34,18,51)(17,48,50,45)(27,56,60,33)(31,62,63,57)(37,59,39,53)(42,61,64,44)(47,54,49,52)

Polycyclic presentation:

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

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#8501, #187508, #187510, …
maximal normal subgroup of	C₂×#8501, #187508, #187510, …
maximal quotient of	#96982, #96984, #96987, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 1489);