#56950

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

627 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)(17,18,21,24)(19,25,27,22)(20,23,28,30)(26,31,32,29)(34,35)(36,43)(37,45)(46,47);(2,6)(7,12)(8,13)(14,16)(20,28)(23,30)(26,32)(29,31)(33,34,38,35)(36,41,43,39)(37,42,45,40)(44,46,48,47);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31)(33,35,38,34)(36,39,43,41)(37,42,45,40)(44,46,48,47);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,36,38,43)(34,39,35,41)(37,44,45,48)(40,46,42,47);(1,4,5,11)(2,8,6,13)(3,10,9,15)(7,14,12,16)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)

Polycyclic presentation:

2^8|a^2=fgh,b^2=f,c^2=gh,d^2=f,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×#56950, #8099759, #8100317, …
maximal normal subgroup of	C₂×#56950, #8099759, #8100317, …
maximal quotient of	#586004, #586863, #587175, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 49938);