#20474

order	256
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	Aut(G) order 32768
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂³
commutator	C₂²	C₂²×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂³	C₂³×C₄
prime radical	C₂⁴	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

473 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^8|a^2=f,b^2=g,c^2=f,d^2=e,f^2=h,b^a=be,d^a=dh

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#191349, #191351, #191359, …
maximal normal subgroup of	#191349, #191351, #191359, …
maximal quotient of	#183724, #183733, #183736, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 13462);