#50928

order	256
nilpotency class	2
isoclinism	isoclinic class of #47826
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

735 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^8|a^2=gh,b^2=g,c^2=g,d^2=h,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#50928, #7706663, #7706670, …
maximal normal subgroup of	C₂×#50928, #7706663, #7706670, …
maximal quotient of	#560360, #560554, #561544, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 43916);