#51307

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

750 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)(11,15)(14,16)(17,18,22,25)(19,26,27,23)(20,24,30,21)(28,32,31,29);(1,3)(2,6)(4,7)(5,8)(9,10,13,12)(11,14,15,16)(19,27)(20,30)(21,24)(23,26);(1,3,5,8)(2,7,4,6)(9,11)(10,14)(12,16)(13,15)(17,19)(18,23)(20,31)(21,29)(22,27)(24,32)(25,26)(28,30);(1,3)(2,7)(4,6)(5,8)(9,10,13,12)(11,14,15,16)(17,20)(18,24)(19,28)(21,25)(22,30)(23,29)(26,32)(27,31);(1,4,5,2)(3,6,8,7)(9,12,13,10)(11,16,15,14)(17,21,22,24)(18,20,25,30)(19,29,27,32)(23,31,26,28)

Polycyclic presentation:

2^8|a^2=fh,b^2=fgh,c^2=f,d^2=fgh,e^2=fg,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×#51307, #7707259, #7707299, …
maximal normal subgroup of	C₂×#51307, #7707259, #7707299, …
maximal quotient of	#560890, #561066, #561994, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 44295);