#19726

order	256
nilpotency class	3
isoclinism	isoclinic class of #19713
automorphism	Aut(G) order 16384
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	#3184	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

362 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#19726, #6357363, #6357368, …
maximal normal subgroup of	C₂×#19726, #6357363, #6357368, …
maximal quotient of	#191524, #206260, #206266, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 12714);