#19718

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	#3207	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

356 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:

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

Polycyclic presentation:

2^8|a^2=g,b^2=e,c^2=fg,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×#19718, #6357363, #6357374, …
maximal normal subgroup of	C₂×#19718, #6357363, #6357374, …
maximal quotient of	#191523, #206259, #206262, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 12706);