#49386

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

738 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)(3,5)(4,8)(6,7)(9,10,13,16)(11,14,12,15)(17,18)(19,21)(20,24)(22,30)(23,26)(25,31)(27,32)(28,29);(1,3)(2,5)(4,6)(7,8)(17,19,23,20)(18,24,26,21)(22,27,31,28)(25,32,30,29);(2,7)(3,4)(9,10)(11,14)(12,15)(13,16)(17,20,23,19)(18,24,26,21)(22,28,31,27)(25,32,30,29);(1,4,6,3)(2,8,7,5)(9,11,13,12)(10,14,16,15)(17,21,23,24)(18,19,26,20)(22,29,31,32)(25,27,30,28);(1,5,6,8)(2,3,7,4)(9,12,13,11)(10,15,16,14)(17,22)(18,25)(19,27)(20,28)(21,29)(23,31)(24,32)(26,30)

Polycyclic presentation:

2^8|a^2=gh,b^2=f,c^2=f,d^2=fg,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×#49386, #7704502, #7704977, …
maximal normal subgroup of	C₂×#49386, #7704502, #7704977, …
maximal quotient of	#560786, #560913, #561885, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 42374);