#57919

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

645 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)(19,24)(20,25)(21,29)(22,23)(26,30)(27,28)(31,32)(33,34,37,40)(35,41,43,38)(36,39,44,46)(42,47,48,45);(4,12)(8,13)(11,15)(14,16)(18,23)(20,24)(26,29)(28,31)(33,34,37,40)(35,38,43,41)(36,39,44,46)(42,45,48,47);(1,2,5,6)(3,10,9,7)(4,8,12,13)(11,16,15,14)(17,19,22,25)(18,24,23,20)(21,27,30,32)(26,31,29,28)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48);(1,3,5,9)(2,7,6,10)(4,11,12,15)(8,14,13,16)(17,20,22,24)(18,25,23,19)(21,28,30,31)(26,32,29,27)(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)(17,21,22,30)(18,26,23,29)(19,27,25,32)(20,28,24,31)(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=fgh,d^2=fg,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×#57919, #8101434, #8101955, …
maximal normal subgroup of	C₂×#57919, #8101434, #8101955, …
maximal quotient of	#586118, #586328, #586470, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 50907);