#54680

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

629 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,6,3)(4,7,11,9)(5,10,13,8)(12,15,16,14)(17,18)(19,23)(20,27)(21,22)(24,28)(25,31)(26,29)(30,32)(34,35)(36,43)(37,45)(46,47);(2,3)(4,11)(5,13)(14,15)(18,22)(23,29)(24,27)(30,31)(33,34,38,35)(36,41,43,39)(37,42,45,40)(44,46,48,47);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,19,21,26)(18,23,22,29)(20,25,28,32)(24,30,27,31)(33,35,38,34)(36,39,43,41)(37,42,45,40)(44,46,48,47);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,29)(27,28)(31,32)(33,36,38,43)(34,39,35,41)(37,44,45,48)(40,46,42,47);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,20,21,28)(18,24,22,27)(19,25,26,32)(23,30,29,31)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)

Polycyclic presentation:

2^8|a^2=h,b^2=g,c^2=f,d^2=g,e^2=fg,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×#54680, #8103643, #8103899, …
maximal normal subgroup of	C₂×#54680, #8103643, #8103899, …
maximal quotient of	#586354, #586946, #587074, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 47668);