#18965

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

Polycyclic presentation:

2^8|b^2=g,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,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×#18965, #6348941, #6348959, …
maximal normal subgroup of	C₂×#18965, #6348941, #6348959, …
maximal quotient of	#170416, #170476, #170509, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 11953);