#57107

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

629 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)(10,15)(17,18,22,25)(19,20,27,23)(21,24,29,31)(26,28,32,30)(34,35)(36,43)(37,45)(46,47);(2,6)(7,12)(8,13)(14,16)(21,29)(24,31)(26,32)(28,30)(33,34,38,35)(36,41,43,39)(37,42,45,40)(44,46,48,47);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,19,22,27)(18,23,25,20)(21,26,29,32)(24,30,31,28)(33,35,38,34)(36,39,43,41)(37,42,45,40)(44,46,48,47);(1,3,5,9)(2,7,6,12)(4,10,11,15)(8,14,13,16)(17,20,22,23)(18,19,25,27)(21,28,29,30)(24,26,31,32)(33,36,38,43)(34,39,35,41)(37,44,45,48)(40,46,42,47);(1,4,5,11)(2,8,6,13)(3,10,9,15)(7,14,12,16)(17,21,22,29)(18,24,25,31)(19,26,27,32)(20,28,23,30)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)

Polycyclic presentation:

2^8|a^2=fgh,b^2=f,c^2=gh,d^2=g,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×#57107, #8099759, #8100305, …
maximal normal subgroup of	C₂×#57107, #8099759, #8100305, …
maximal quotient of	#586487, #586798, #587007, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 50095);