#58680

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

643 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:

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

Polycyclic presentation:

2^8|a^2=fg,b^2=g,c^2=fgh,d^2=fh,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×#58680, #8104855, #8104904, …
maximal normal subgroup of	C₂×#58680, #8104855, #8104904, …
maximal quotient of	#586422, #586525, #587296, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 51668);