#41685

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	C₂⁴	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

672 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:

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

Polycyclic presentation:

2^8|a^2=fgh,b^2=g,c^2=f,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×#41685, #7640139, #7640240, …
maximal normal subgroup of	C₂×#41685, #7640139, #7640240, …
maximal quotient of	#544753, #545241, #545799, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 34673);