#57742

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

631 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)(17,18,22,19)(20,23,27,25)(21,26,29,24)(28,31,32,30)(33,34,39,36)(35,37,40,43)(38,41,47,44)(42,45,46,48);(4,12)(8,13)(11,15)(14,16)(18,19)(20,27)(21,29)(30,31)(33,35,39,40)(34,37,36,43)(38,46,47,42)(41,48,44,45);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,19,22,18)(20,23,27,25)(21,26,29,24)(28,30,32,31)(33,36,39,34)(35,37,40,43)(38,44,47,41)(42,45,46,48);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,15)(13,16)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,37)(34,40)(35,36)(38,45)(39,43)(41,46)(42,44)(47,48);(1,4,5,12)(2,8,6,13)(3,11,9,15)(7,14,10,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,38)(34,41)(35,42)(36,44)(37,45)(39,47)(40,46)(43,48)

Polycyclic presentation:

2^8|a^2=h,b^2=fh,c^2=h,d^2=f,e^2=g,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×#57742, #8101754, #8101961, …
maximal normal subgroup of	C₂×#57742, #8101754, #8101961, …
maximal quotient of	#586881, #587000, #587230, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 50730);