#56166

order	256
nilpotency class	2
isoclinism	isoclinic class of #53390
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	C₂⁴	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

629 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)(3,7)(4,11)(5,6)(8,12)(9,15)(10,13)(14,16)(17,18,22,21)(19,25,26,23)(20,28,29,24)(27,30,32,31)(34,35)(36,43)(37,45)(46,47);(2,6)(7,13)(8,11)(14,15)(19,26)(23,25)(27,32)(30,31)(33,34,38,35)(36,41,43,39)(37,42,45,40)(44,46,48,47);(1,3,5,10)(2,7,6,13)(4,9,12,16)(8,14,11,15)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,35,38,34)(36,39,43,41)(37,42,45,40)(44,46,48,47);(1,2)(3,7)(4,8)(5,6)(9,14)(10,13)(11,12)(15,16)(17,20,22,29)(18,24,21,28)(19,27,26,32)(23,30,25,31)(33,36,38,43)(34,39,35,41)(37,44,45,48)(40,46,42,47);(1,4)(2,8)(3,9)(5,12)(6,11)(7,14)(10,16)(13,15)(17,21,22,18)(19,25,26,23)(20,28,29,24)(27,31,32,30)(33,37,38,45)(34,40,35,42)(36,44,43,48)(39,46,41,47)

Polycyclic presentation:

2^8|a^2=h,b^2=gh,c^2=f,d^2=g,e^2=g,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#56166, #8103020, #8103194, …
maximal normal subgroup of	C₂×#56166, #8103020, #8103194, …
maximal quotient of	#586530, #586974, #587072, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 49154);