#58450

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

643 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:

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

Polycyclic presentation:

2^8|a^2=h,b^2=h,c^2=f,d^2=h,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×#58450, #8104264, #8104431, …
maximal normal subgroup of	C₂×#58450, #8104264, #8104431, …
maximal quotient of	#586356, #586637, #586925, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 51438);