#41709

order	256
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	Aut(G) = C₂¹⁵ (order 32768) · Inn(G) = C₂⁵ · Out(G) = C₂¹⁰
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁵
prime radical	C₂²×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

697 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^8|a^2=fh,c^2=fg,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#41709, #7640139, #7640237, …
maximal normal subgroup of	C₂×#41709, #7640139, #7640237, …
maximal quotient of	#544859, #545284, #545821, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 34697);