#56870

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

Polycyclic presentation:

2^8|a^2=fh,b^2=g,c^2=fh,d^2=f,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×#56870, #8099854, #8100312, …
maximal normal subgroup of	C₂×#56870, #8099854, #8100312, …
maximal quotient of	#585907, #586967, #587281, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 49858);