#18796

order	256
nilpotency class	3
isoclinism	isoclinic class of #18746
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

473 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18)(19,27)(20,23)(21,26)(22,25)(24,29)(28,32)(30,31);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(18,23)(20,21)(24,31)(25,26)(27,32)(28,29);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,19,22,30)(18,24,25,32)(20,29,21,28)(23,31,26,27);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(18,25)(23,26)(24,32)(27,31)

Polycyclic presentation:

2^8|c^2=g,d^2=gh,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,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×#18796, #6348715, #6348737, …
maximal normal subgroup of	C₂×#18796, #6348715, #6348737, …
maximal quotient of	#166159, #166167, #166175, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 11784);