#20340

order	256
nilpotency class	2
isoclinism	isoclinic class of #20338
automorphism	Aut(G) order 589824
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂	C₂²×#95
prime radical	C₂²×#41	C₂²
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

978 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#11709427
Fitting subgroup of	#11709426, #11709427, #11709428
maximal subgroup of	#138264, #138265, #356479, …
maximal normal subgroup of	#138264, #138265, #356479, …
maximal quotient of	#180877, #180881, #180885, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 13328);