#52856

order	256
nilpotency class	2
isoclinism	isoclinic class of #52308
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

647 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,6,4)(3,9,11,7)(5,8,13,12)(10,15,16,14)(17,18)(19,24)(20,28)(21,26)(22,23)(25,29)(27,30)(31,32)(36,44)(39,46)(42,48)(45,47);(2,4)(7,9)(8,12)(14,15)(18,23)(19,26)(20,29)(30,31)(33,34,37,40)(35,38,43,41)(36,39,44,46)(42,45,48,47);(1,3)(2,7)(4,9)(5,10)(6,11)(8,14)(12,15)(13,16)(17,19,22,26)(18,24,23,21)(20,27,29,32)(25,31,28,30)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(1,4,6,2)(3,9,11,7)(5,12,13,8)(10,15,16,14)(17,20)(18,25)(19,27)(21,30)(22,29)(23,28)(24,31)(26,32)(33,35)(34,38)(36,42)(37,43)(39,45)(40,41)(44,48)(46,47);(1,5,6,13)(2,8,4,12)(3,10,11,16)(7,14,9,15)(17,21,22,24)(18,26,23,19)(20,30,29,31)(25,32,28,27)(33,36)(34,39)(35,42)(37,44)(38,45)(40,46)(41,47)(43,48)

Polycyclic presentation:

2^8|a^2=g,b^2=fgh,c^2=fg,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dfg,e^a=efgh,c^b=ch,d^b=df,d^c=dfgh

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#52856, #8092679, #8092704, …
maximal normal subgroup of	C₂×#52856, #8092679, #8092704, …
maximal quotient of	#582335, #582512, #582557, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 45844);