#18726

order	256
nilpotency class	3
isoclinism	isoclinic class of #18712
automorphism	Aut(G) order 4096
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	C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

446 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#18726, #6348342, #6348358, …
maximal normal subgroup of	C₂×#18726, #6348342, #6348358, …
maximal quotient of	#168127, #168166, #168181, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 11714);