#19711

order	256
nilpotency class	3
isoclinism	isoclinic class of #19697
automorphism	Aut(G) order 65536
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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#143193, #143195, #143665, …
maximal normal subgroup of	#143193, #143195, #143665, …
maximal quotient of	#191457, #206208, #206250, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 12699);