#46221

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	C₂²×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

697 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:

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

Polycyclic presentation:

2^8|a^2=fh,c^2=fg,d^2=fgh,e^2=g,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×#46221, #7702765, #7702852, …
maximal normal subgroup of	C₂×#46221, #7702765, #7702852, …
maximal quotient of	#557180, #557220, #557474, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 39209);