#49856

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

785 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:

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

Polycyclic presentation:

2^8|a^2=fg,b^2=f,c^2=h,d^2=f,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×#49856, #7705619, #7705675, …
maximal normal subgroup of	C₂×#49856, #7705619, #7705675, …
maximal quotient of	#560712, #561143, #561815, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 42844);