#54689

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:

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

Polycyclic presentation:

2^8|a^2=fh,b^2=g,c^2=f,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×#54689, #8101621, #8101959, …
maximal normal subgroup of	C₂×#54689, #8101621, #8101959, …
maximal quotient of	#586330, #586624, #587070, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 47677);