#57770

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:

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

Polycyclic presentation:

2^8|a^2=fh,b^2=fgh,c^2=h,d^2=f,e^2=fg,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×#57770, #8101488, #8101963, …
maximal normal subgroup of	C₂×#57770, #8101488, #8101963, …
maximal quotient of	#586366, #586597, #586916, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 50758);