#58400

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	C₂²×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

694 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:

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

Polycyclic presentation:

2^8|a^2=gh,b^2=h,c^2=fg,d^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×#58400, #8104276, #8104424, …
maximal normal subgroup of	C₂×#58400, #8104276, #8104424, …
maximal quotient of	#585492, #585722, #586924, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 51388);