#26651

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
prime radical	C₂×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

383 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10,13,19,14,20,15,21)(11,22,25,33,26,34,27,16)(12,28,29,36,30,17,31,18)(23,37,24,32,38,35,39,40);(9,11,14,26)(10,16,20,33)(12,24,30,39)(13,27,15,25)(17,35,28,37)(18,32,36,40)(19,34,21,22)(23,31,38,29);(2,5)(6,8)(9,12)(10,17)(11,23)(13,29)(14,30)(15,31)(16,32)(18,19)(20,28)(21,36)(22,35)(24,25)(26,38)(27,39)(33,40)(34,37);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,24)(13,31)(14,30)(15,29)(16,32)(17,19)(20,36)(21,28)(22,37)(23,25)(26,39)(27,38)(33,40)(34,35)

Polycyclic presentation:

2^8|a^2=e,b^2=h,d^2=fh,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,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×#26651, #6439605, #6439607, …
maximal normal subgroup of	C₂×#26651, #6439605, #6439607, …
maximal quotient of	#309994, #309995, #309997, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 19639);