#24461

order	256
nilpotency class	3
isoclinism	isoclinic class of #23886
automorphism	Aut(G) order 4096
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

659 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)(7,16)(8,22)(9,25)(10,14)(11,15)(12,19)(13,23)(17,21)(18,27)(20,28)(24,29)(26,30)(31,33)(32,35)(34,36);(5,7,11,21)(6,12,15,28)(8,18,24,34)(9,20,10,19)(13,27,30,36)(14,17,25,16)(22,31,29,32)(23,35,26,33);(2,4)(5,6)(7,17)(8,13)(9,14)(10,25)(11,15)(12,19)(16,21)(18,32)(20,28)(22,23)(24,30)(26,29)(27,35)(31,34)(33,36);(5,8,11,24)(6,13,15,30)(7,18,21,34)(9,26,10,23)(12,27,28,36)(14,29,25,22)(16,31,17,32)(19,35,20,33)

Polycyclic presentation:

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

Conjugacy classes & character table

43 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#24461, #6437902, #6437906, …
maximal normal subgroup of	C₂×#24461, #6437902, #6437906, …
maximal quotient of	#268634, #268637, #268648, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 17449);