#51178

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

743 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:

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

Polycyclic presentation:

2^8|b^2=fgh,c^2=f,d^2=gh,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×#51178, #7707136, #7707185, …
maximal normal subgroup of	C₂×#51178, #7707136, #7707185, …
maximal quotient of	#560780, #561082, #561900, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 44166);