#21248

order	256
nilpotency class	3
isoclinism	isoclinic class of #471
automorphism	Aut(G) order 32768
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×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×#135	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

729 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#147416, #147419, #147434, …
maximal normal subgroup of	#147416, #147419, #147434, …
maximal quotient of	#204429, #204435, #204438, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 14236);