#58476

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

631 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)(10,15)(17,18)(19,25)(20,27)(21,29)(22,23)(24,30)(26,28)(31,32)(33,34,39,41)(35,42,36,37)(38,46,47,40)(43,45,44,48);(2,6)(7,12)(8,13)(14,16)(18,23)(20,25)(21,30)(26,32)(33,35,39,36)(34,37,41,42)(38,43,47,44)(40,45,46,48);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,19,22,27)(18,20,23,25)(21,26,30,32)(24,28,29,31)(33,36,39,35)(34,37,41,42)(38,44,47,43)(40,45,46,48);(1,3,5,9)(2,7,6,12)(4,10,11,15)(8,14,13,16)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,32)(30,31)(33,37)(34,35)(36,41)(38,45)(39,42)(40,43)(44,46)(47,48);(1,4,5,11)(2,8,6,13)(3,10,9,15)(7,14,12,16)(17,21)(18,24)(19,26)(20,28)(22,30)(23,29)(25,31)(27,32)(33,38,39,47)(34,40,41,46)(35,43,36,44)(37,45,42,48)

Polycyclic presentation:

2^8|a^2=fgh,b^2=fgh,c^2=f,d^2=h,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×#58476, #8104264, #8104432, …
maximal normal subgroup of	C₂×#58476, #8104264, #8104432, …
maximal quotient of	#586484, #586621, #586685, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 51464);