#58789

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

Polycyclic presentation:

2^8|a^2=fg,b^2=gh,c^2=fgh,d^2=fh,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×#58789, #8104949, #8105006, …
maximal normal subgroup of	C₂×#58789, #8104949, #8105006, …
maximal quotient of	#586357, #586469, #587296, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 51777);