#49498

order	256
nilpotency class	2
isoclinism	isoclinic class of #47826
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	C₂×#121	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

717 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:

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

Polycyclic presentation:

2^8|a^2=h,b^2=fg,c^2=fg,d^2=g,e^2=g,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#49498, #7704282, #7704982, …
maximal normal subgroup of	C₂×#49498, #7704282, #7704982, …
maximal quotient of	#560754, #561046, #561841, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 42486);