#58778

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

627 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)(3,6)(4,9)(5,10)(7,11)(8,13)(12,14)(15,16)(17,18,21,24)(19,25,27,22)(20,23,28,30)(26,31,32,29)(33,34,37,40)(35,41,43,38)(36,39,44,46)(42,47,48,45);(2,3)(4,11)(8,10)(12,16)(20,28)(23,30)(26,32)(29,31)(33,34,37,40)(35,38,43,41)(36,39,44,46)(42,45,48,47);(1,3,6,2)(4,9,11,7)(5,10,13,8)(12,15,16,14)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)(33,36,37,44)(34,39,40,46)(35,42,43,48)(38,45,41,47)

Polycyclic presentation:

2^8|a^2=fg,b^2=fh,c^2=fgh,d^2=fh,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×#58778, #8104949, #8104959, …
maximal normal subgroup of	C₂×#58778, #8104949, #8104959, …
maximal quotient of	#586150, #586342, #587136, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 51766);