#57117

order	256
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	Aut(G) order 98304
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	#392	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

669 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:

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

Fitting subgroup of	#57117⋊C₃
maximal subgroup of	C₂×#57117, #8100102, #8100327, …
maximal normal subgroup of	C₂×#57117, #8100102, #8100327, …
maximal quotient of	#586407, #588632, #591721, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 50105);