#49521

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	C₂×#121	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

763 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:

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

Polycyclic presentation:

2^8|a^2=fgh,b^2=g,c^2=fg,d^2=g,e^2=g,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×#49521, #7704237, #7704983, …
maximal normal subgroup of	C₂×#49521, #7704237, #7704983, …
maximal quotient of	#560675, #560917, #561775, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 42509);