#46251

order	256
nilpotency class	2
isoclinism	isoclinic class of #45438
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

676 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#46251, #7702877, #7702895, …
maximal normal subgroup of	C₂×#46251, #7702877, #7702895, …
maximal quotient of	#557182, #557203, #557489, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 39239);