#57731

order	256
nilpotency class	2
isoclinism	isoclinic class of #53390
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

671 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)(3,7)(9,10,14,18)(11,16,12,15)(13,21,22,17)(19,23,20,24)(25,26,31,34)(27,32,28,29)(30,33,38,40)(35,39,36,37);(2,5)(6,8)(9,11,14,12)(10,15,18,16)(13,20,22,19)(17,24,21,23)(25,27,31,28)(26,32,34,29)(30,36,38,35)(33,37,40,39);(1,2)(3,6)(4,5)(7,8)(9,12,14,11)(10,16,18,15)(13,20,22,19)(17,24,21,23)(25,28,31,27)(26,32,34,29)(30,36,38,35)(33,39,40,37);(1,3,4,7)(2,6,5,8)(9,10)(11,16)(12,15)(13,17)(14,18)(19,24)(20,23)(21,22)(25,29,31,32)(26,28,34,27)(30,37,38,39)(33,36,40,35);(9,13,14,22)(10,17,18,21)(11,19,12,20)(15,23,16,24)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)

Polycyclic presentation:

2^8|a^2=gh,b^2=gh,c^2=gh,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#57731, #8101914, #8101952, …
maximal normal subgroup of	C₂×#57731, #8101914, #8101952, …
maximal quotient of	#586296, #586866, #586910, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 50719);