#20678

order	256
nilpotency class	3
isoclinism	isoclinic class of #471
automorphism	Aut(G) order 32768
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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

665 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)(3,4)(5,6,10,15,11,16,9,14)(7,17,20,28,21,27,19,12)(8,22,23,13,25,29,24,30)(18,32,33,31,35,26,34,36);(5,7,9,19,11,21,10,20)(6,12,15,17,16,28,14,27)(8,18,23,33,25,35,24,34)(13,26,22,31,30,32,29,36);(2,4)(5,8)(6,13)(7,18)(9,23)(10,24)(11,25)(12,26)(14,29)(15,22)(16,30)(17,31)(19,33)(20,34)(21,35)(27,36)(28,32);(5,9,11,10)(6,14,16,15)(7,19,21,20)(8,23,25,24)(12,27,28,17)(13,29,30,22)(18,33,35,34)(26,36,32,31)

Polycyclic presentation:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#146437, #146440, #146454, …
maximal normal subgroup of	#146437, #146440, #146454, …
maximal quotient of	#188780, #188784, #188786, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 13666);