#46840

order	256
nilpotency class	2
isoclinism	isoclinic class of #46350
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

708 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)(5,8)(9,10,14,11)(12,15,19,17)(13,18,21,16)(20,23,24,22)(25,26)(27,32)(28,33)(29,37)(30,31)(34,38)(35,36)(39,40);(1,2,4,6)(3,5,7,8)(10,11)(12,19)(13,21)(22,23)(26,31)(28,32)(34,37)(36,39);(1,3)(2,5)(4,7)(6,8)(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23)(25,27,30,33)(26,32,31,28)(29,35,38,40)(34,39,37,36);(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,28,30,32)(26,33,31,27)(29,36,38,39)(34,40,37,35);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29)(26,34)(27,35)(28,36)(30,38)(31,37)(32,39)(33,40)

Polycyclic presentation:

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

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#46840, #7700704, #7700720, …
maximal normal subgroup of	C₂×#46840, #7700704, #7700720, …
maximal quotient of	#576622, #576626, #576637, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 39828);