#58402

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

673 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)(3,7)(4,5)(6,8)(9,10,14,15)(11,12,19,16)(13,21,22,17)(18,23,24,20)(25,26,30,33)(27,31,36,40)(28,29,32,39)(34,38,37,35);(2,5)(6,7)(10,15)(12,16)(13,22)(18,24)(25,27,30,36)(26,31,33,40)(28,34,32,37)(29,38,39,35);(9,11,14,19)(10,16,15,12)(13,18,22,24)(17,23,21,20)(25,27)(26,31)(28,37)(29,35)(30,36)(32,34)(33,40)(38,39);(1,2)(3,6)(4,5)(7,8)(9,12,14,16)(10,11,15,19)(13,20,22,23)(17,18,21,24)(25,28,30,32)(26,29,33,39)(27,34,36,37)(31,38,40,35);(1,3,4,8)(2,6,5,7)(9,13,14,22)(10,17,15,21)(11,18,19,24)(12,20,16,23)(25,29)(26,32)(27,35)(28,33)(30,39)(31,34)(36,38)(37,40)

Polycyclic presentation:

2^8|a^2=gh,b^2=h,c^2=g,d^2=gh,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×#58402, #8104369, #8104424, …
maximal normal subgroup of	C₂×#58402, #8104369, #8104424, …
maximal quotient of	#585778, #586160, #586660, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 51390);