#47530

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

692 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)(9,10,14,17)(11,13,16,22)(12,15,21,24)(18,20,19,23)(26,31)(27,34)(29,37)(36,39);(11,16)(13,22)(18,19)(20,23)(25,26,30,31)(27,28,34,32)(29,38,37,33)(35,39,40,36);(1,2)(3,6)(4,5)(7,8)(9,11,14,16)(10,13,17,22)(12,19,21,18)(15,23,24,20)(25,27,30,34)(26,32,31,28)(29,35,37,40)(33,39,38,36);(9,12)(10,15)(11,18)(13,20)(14,21)(16,19)(17,24)(22,23)(25,28,30,32)(26,27,31,34)(29,36,37,39)(33,35,38,40);(1,3,4,7)(2,6,5,8)(9,13)(10,16)(11,17)(12,20)(14,22)(15,19)(18,24)(21,23)(25,29,30,37)(26,33,31,38)(27,35,34,40)(28,36,32,39)

Polycyclic presentation:

2^8|a^2=fh,b^2=f,c^2=h,d^2=f,e^2=g,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×#47530, #7700931, #7700944, …
maximal normal subgroup of	C₂×#47530, #7700931, #7700944, …
maximal quotient of	#576984, #577000, #577114, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 40518);