#18945

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
prime radical	#3552	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

423 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:

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

Polycyclic presentation:

2^8|a^2=g,b^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×#18945, #6348928, #6348958, …
maximal normal subgroup of	C₂×#18945, #6348928, #6348958, …
maximal quotient of	#170396, #170444, #170480, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 11933);