#56249

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

Polycyclic presentation:

2^8|a^2=fh,b^2=gh,c^2=fg,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×#56249, #8099917, #8100308, …
maximal normal subgroup of	C₂×#56249, #8099917, #8100308, …
maximal quotient of	#586689, #586759, #587282, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 49237);