#21075

order	256
nilpotency class	3
isoclinism	isoclinic class of #471
automorphism	Aut(G) order 16384
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#121
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂²	C₂×#121
prime radical	C₂²×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

507 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2,4,3)(5,6)(7,16)(8,21)(9,14)(10,25)(11,15)(12,19)(13,24)(17,31)(18,27)(20,30)(22,36)(23,28)(26,32)(29,34)(33,35);(1,3,4,2)(5,7,10,19,11,20,9,18)(6,12,14,16,15,27,25,30)(8,17,23,33,24,34,22,32)(13,26,28,29,21,35,36,31);(5,8,10,23,11,24,9,22)(6,13,14,28,15,21,25,36)(7,17,19,33,20,34,18,32)(12,26,16,29,27,35,30,31);(5,9,11,10)(6,14,15,25)(7,18,20,19)(8,22,24,23)(12,16,27,30)(13,28,21,36)(17,32,34,33)(26,29,35,31)

Polycyclic presentation:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#354399, #354400, #357150, …
maximal normal subgroup of	#354399, #354400, #357150, …
maximal quotient of	#202478, #202484, #202490, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 14063);