#56786

order	256
nilpotency class	2
isoclinism	isoclinic class of #53390
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

685 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)(3,7)(9,10,14,17)(11,12,19,15)(13,16,21,23)(18,20,24,22)(25,26)(27,29)(28,32)(30,38)(31,34)(33,39)(35,40)(36,37);(2,5)(6,8)(13,21)(16,23)(18,24)(20,22)(25,27,31,28)(26,32,34,29)(30,35,39,36)(33,40,38,37);(1,2)(3,6)(4,5)(7,8)(9,11,14,19)(10,15,17,12)(13,18,21,24)(16,22,23,20)(25,28,31,27)(26,32,34,29)(30,36,39,35)(33,40,38,37);(1,3,4,7)(2,6,5,8)(9,12,14,15)(10,11,17,19)(13,20,21,22)(16,18,23,24)(25,29,31,32)(26,27,34,28)(30,37,39,40)(33,35,38,36);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,30,31,39)(26,33,34,38)(27,35,28,36)(29,37,32,40)

Polycyclic presentation:

2^8|a^2=gh,b^2=f,c^2=fgh,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#56786, #8100133, #8100302, …
maximal normal subgroup of	C₂×#56786, #8100133, #8100302, …
maximal quotient of	#586256, #586263, #587261, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 49774);