#18891

order	256
nilpotency class	3
isoclinism	isoclinic class of #18866
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

450 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)(3,9)(4,12)(5,7)(6,11)(8,14)(10,16)(13,15)(19,26)(20,21)(23,30)(24,28)(27,29)(31,32);(1,3,5,10)(2,6,7,15)(4,13,14,11)(8,16,12,9)(17,18)(19,23)(20,24)(21,28)(22,25)(26,31)(27,32)(29,30);(17,19)(18,23)(20,26)(21,29)(22,27)(24,31)(25,32)(28,30);(2,7)(3,10)(8,12)(11,13)(17,20,22,21)(18,24,25,28)(19,26,27,29)(23,31,32,30)

Polycyclic presentation:

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

Conjugacy classes & character table

43 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#18891, #6348854, #6348888, …
maximal normal subgroup of	C₂×#18891, #6348854, #6348888, …
maximal quotient of	#167517, #167534, #167551, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 11879);