#13829

order	256
nilpotency class	2
isoclinism	isoclinic class of #560
automorphism	Aut(G) order 1572864
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₄⁴

Aliases

No aliases for this group.

Subgroups

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

663 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^8|a^2=g,b^2=h,c^2=f,d^2=e,b^a=be,c^a=cf,d^a=def,c^b=cef,d^b=de

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#13829⋊C₃, #72240345, C₂×#13829⋊C₃, …
Fitting subgroup of	#13829⋊C₃, #72260904
maximal subgroup of	#109999, #110000, #110105, …
maximal normal subgroup of	#109999, #110000, #110105, …
maximal quotient of	#153731, #153957, #153960, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6817);