#58914

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

669 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:

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

Polycyclic presentation:

2^8|a^2=fh,b^2=g,c^2=fg,d^2=f,e^2=fgh,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×#58914, #8105321, #8105330, …
maximal normal subgroup of	C₂×#58914, #8105321, #8105330, …
maximal quotient of	#585749, #585957, #586147, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 51902);