#54260

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

627 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:

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

Polycyclic presentation:

2^8|a^2=fh,b^2=f,c^2=g,d^2=fg,e^2=f,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×#54260, #8101595, #8101966, …
maximal normal subgroup of	C₂×#54260, #8101595, #8101966, …
maximal quotient of	#586014, #586866, #587192, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 47248);