#57115

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

625 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)(17,18,22,25)(19,20,27,23)(21,24,29,31)(26,28,32,30)(33,34)(35,37)(36,40)(38,46)(39,42)(41,47)(43,48)(44,45);(2,6)(7,12)(8,13)(14,16)(21,29)(24,31)(26,32)(28,30)(33,35,39,36)(34,40,42,37)(38,43,47,44)(41,48,46,45);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,19,22,27)(18,23,25,20)(21,26,29,32)(24,30,31,28)(33,36,39,35)(34,40,42,37)(38,44,47,43)(41,48,46,45);(1,3,5,9)(2,7,6,12)(4,10,11,15)(8,14,13,16)(17,20,22,23)(18,19,25,27)(21,28,29,30)(24,26,31,32)(33,37,39,40)(34,35,42,36)(38,45,47,48)(41,43,46,44);(1,4,5,11)(2,8,6,13)(3,10,9,15)(7,14,12,16)(17,21,22,29)(18,24,25,31)(19,26,27,32)(20,28,23,30)(33,38,39,47)(34,41,42,46)(35,43,36,44)(37,45,40,48)

Polycyclic presentation:

2^8|a^2=gh,b^2=f,c^2=fgh,d^2=fg,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×#57115, #8099569, #8100305, …
maximal normal subgroup of	C₂×#57115, #8099569, #8100305, …
maximal quotient of	#586254, #586455, #587263, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 50103);