#19736

order	256
nilpotency class	3
isoclinism	isoclinic class of #19733
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	C₂³	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

318 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(3,9)(4,11)(6,12)(8,14)(10,15)(13,16)(17,18,22,26)(19,27,30,41)(20,24,32,38)(21,34,28,25)(23,35,37,31)(29,40,44,48)(33,45,42,39)(36,46,47,43);(1,2)(3,6)(4,8)(5,7)(9,12)(10,13)(11,14)(15,16)(17,19,28,35,22,30,21,31)(18,23,34,41,26,37,25,27)(20,29,42,46,32,44,33,43)(24,36,45,48,38,47,39,40);(1,3,4,9,5,10,11,15)(2,6,8,12,7,13,14,16)(17,20)(18,24)(19,29)(21,33)(22,32)(23,36)(25,39)(26,38)(27,40)(28,42)(30,44)(31,43)(34,45)(35,46)(37,47)(41,48);(2,7)(6,13)(8,14)(12,16)(19,30)(20,32)(23,37)(24,38)(27,41)(31,35)(33,42)(39,45)

Polycyclic presentation:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#19736, #6357385, #6357389, …
maximal normal subgroup of	C₂×#19736, #6357385, #6357389, …
maximal quotient of	#191528, #206275, #206279, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 12724);