#57081

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

645 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)(11,15)(17,18,22,21)(19,25,26,23)(20,28,29,24)(27,30,32,31)(33,34)(35,37)(36,40)(38,46)(39,42)(41,47)(43,48)(44,45);(4,12)(8,13)(11,15)(14,16)(19,26)(23,25)(27,32)(30,31)(33,35,39,36)(34,40,42,37)(38,43,47,44)(41,48,46,45);(1,2,5,6)(3,10,9,7)(4,8,12,13)(11,16,15,14)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,36,39,35)(34,40,42,37)(38,44,47,43)(41,48,46,45);(1,3,5,9)(2,7,6,10)(4,11,12,15)(8,14,13,16)(17,20,22,29)(18,24,21,28)(19,27,26,32)(23,30,25,31)(33,37,39,40)(34,35,42,36)(38,45,47,48)(41,43,46,44);(1,4,5,12)(2,8,6,13)(3,11,9,15)(7,14,10,16)(17,21,22,18)(19,25,26,23)(20,28,29,24)(27,31,32,30)(33,38)(34,41)(35,43)(36,44)(37,45)(39,47)(40,48)(42,46)

Polycyclic presentation:

2^8|a^2=h,b^2=f,c^2=fgh,d^2=fg,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×#57081, #8100163, #8100304, …
maximal normal subgroup of	C₂×#57081, #8100163, #8100304, …
maximal quotient of	#586101, #586413, #586543, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 50069);