#18963

order	256
nilpotency class	3
isoclinism	isoclinic class of #18930
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
prime radical	#3247	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

422 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)(19,26)(20,21)(23,30)(24,28)(27,29)(31,32);(1,3,6,12)(2,7,9,16)(4,11,5,14)(8,15,13,10)(17,18)(19,23)(20,24)(21,28)(22,25)(26,31)(27,32)(29,30);(1,2)(3,7)(4,13)(5,8)(6,9)(10,11)(12,16)(14,15)(17,19)(18,23)(20,26)(21,29)(22,27)(24,31)(25,32)(28,30);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(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=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,d^c=dg,e^c=eh,f^c=fh

Conjugacy classes & character table

37 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#18963, #6348941, #6348960, …
maximal normal subgroup of	C₂×#18963, #6348941, #6348960, …
maximal quotient of	#170414, #170474, #170511, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 11951);