#54583

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

652 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:

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

Polycyclic presentation:

2^8|a^2=fgh,b^2=g,c^2=fg,d^2=f,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×#54583, #8102924, #8103186, …
maximal normal subgroup of	C₂×#54583, #8102924, #8103186, …
maximal quotient of	#585917, #586970, #587298, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 47571);