#58401

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

696 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)(6,7)(9,10,14,12)(11,15,19,17)(13,20,21,16)(18,23,24,22)(25,26,30,33)(27,28,35,31)(29,32,37,39)(34,36,40,38);(1,2,4,5)(3,7,8,6)(10,12)(15,17)(16,20)(22,23)(29,37)(32,39)(34,40)(36,38);(1,3,4,8)(2,6,5,7)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,27,30,35)(26,31,33,28)(29,34,37,40)(32,38,39,36);(1,3)(2,6)(4,8)(5,7)(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,28,30,31)(26,27,33,35)(29,36,37,38)(32,34,39,40);(1,2)(3,6)(4,5)(7,8)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)

Polycyclic presentation:

2^8|a^2=fgh,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×#58401, #8104286, #8104336, …
maximal normal subgroup of	C₂×#58401, #8104286, #8104336, …
maximal quotient of	#585722, #585908, #586532, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 51389);