#53370

order	256
nilpotency class	2
isoclinism	isoclinic class of #53158
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

631 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:

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

Polycyclic presentation:

2^8|a^2=f,b^2=g,c^2=fgh,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=df,e^b=egh

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#53370, #8105910, #8105920, …
maximal normal subgroup of	C₂×#53370, #8105910, #8105920, …
maximal quotient of	#606832, #606841, #606847, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 46358);