#58482

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

633 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:

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

Polycyclic presentation:

2^8|a^2=fgh,b^2=fh,c^2=fg,d^2=h,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×#58482, #8104319, #8104435, …
maximal normal subgroup of	C₂×#58482, #8104319, #8104435, …
maximal quotient of	#585904, #586113, #586764, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 51470);