#55772

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

689 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^8|a^2=fgh,b^2=fh,c^2=f,d^2=g,e^2=f,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×#55772, #8102975, #8103020, …
maximal normal subgroup of	C₂×#55772, #8102975, #8103020, …
maximal quotient of	#585929, #586219, #586960, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 48760);