#12305

order	256
nilpotency class	4
isoclinism	isoclinic class of #12304
automorphism	Aut(G) = #72308439 (order 2048) · Inn(G) = #1616 · Out(G) = C₂²×C₄
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#1616
commutator, Frattini, cosocle	C₂×M₁₆	C₂³
prime radical	C₂³	C₂×SD₁₆
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

131 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,6,26,7,12,28,11)(3,13,17,35,16,40,27,41)(4,18,21,34,20,9,32,48)(5,22,25,10,24,54,55,37)(8,23,38,15,31,45,30,46)(14,39,43,63,44,57,53,60)(19,49,51,36,52,61,59,33)(29,56,62,64,47,50,58,42);(1,3,7,16)(2,8,12,31)(4,14,20,44)(5,23,24,45)(6,27,28,17)(9,29,18,47)(10,35,37,41)(11,38,26,30)(13,22,40,54)(15,25,46,55)(19,50,52,56)(21,53,32,43)(33,60,36,63)(34,62,48,58)(39,49,57,61)(42,51,64,59);(1,4)(2,9,12,18)(3,14)(5,19)(6,21)(7,20)(8,29,31,47)(10,36,37,33)(11,34,26,48)(13,39,40,57)(15,42)(16,44)(17,43)(22,49,54,61)(23,50)(24,52)(25,51)(27,53)(28,32)(30,58,38,62)(35,63,41,60)(45,56)(46,64)(55,59)

Polycyclic presentation:

2^8|a^2=fg,b^2=h,c^2=g,d^2=f,f^2=h,b^a=bd,c^a=ce,d^a=df,f^a=fh,g^a=gh,d^b=dfh,e^b=egh,f^b=fh,d^c=dg,e^c=eh,e^d=eh

Conjugacy classes & character table

25 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#12305, #484982, #484984, …
maximal normal subgroup of	C₂×#12305, #484982, #484984, …
maximal quotient of	#126354, #126392, #126451, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 5293);